Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich
31 Symposium on Probability Methods "Analysis in Lectures delivered at a symposium at Loutraki, Greece, 22. 5. - 4. 6. 1966 Chairman: Professor D. A. Kappos
1967
Springer-Verlag. Berlin. Heidelberg-New York
This symposium was supported by the Scientific Affairs Divisions of the North Atlantic Treaty Organization.
All rights, especiallythat of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomech~mJcal means (photostat, micro~Jm and/or microcard)or by other procedure without wr/tten ~ s / o n from Springer Verlag. @ by Springer-Verhg Berlin. Heidelberg 1967. Library of C O ~ s s C~dog Card Number 67-16670. Printed in Germany. T/de No.7351.
Contents Remarques sur les Th@or~mes de Bochner et P. Levy (A. Badrikian)
I
Recent developments in Axiomatic Potential Theory (H. Bauer)
20
8ber wesentlich indefinite Spiele (D. Bierlein)
28
La Topologie fine en Th@orie du Potential (M. Brelot)
36
Lois Stables et Espaces L p (J. Bretagnolle, D. DacnnbaCastelle, et J. L. Krivine) @8 Comments on the Martingale Convergence Theorem (S. D. Chatterji)
55
Faktorisierung von Differentialoperationen (H. Dinges)
62
Die Anzahl der Niveaudurchg~ge und der lokalen Maximalstellen yon Gau~schen Prozessen (W. Fieger) 63 Vektorwertige Masse und Zufallsvariable auf Boolesche Algebren und der Satz yon Radon-Nikodym (P. Georgiou)
69
Toward a Theory of Patterns (U. Grenander)
79
On the Potential Theory of Linear, Homogeneous Parabolic Partial Differential Equations of Second Order (S.Guber) 112 Invariant and non-invariant Measures (K. Jacobs)
118
Representation of Abstract L-Spaces (D. A. Kappos)
136
Extensions of stationary Processes (H. G. Kellerer)
146
Renewal Sequences and their Arithmetic (D. G. Kendall) 147 Optimal Bounded Control with Linear Stochastic Equations and Quadratic Cost (E. Kounias) 176
On a Fourier Transform in Infinitely many Dimensions (P. Kristensen, L. Mejlbo, and E.T. Poulsen)
187
Some Problems Arising from Spectral Analysis (R. M. Loynes)
197
Analytical Methods in Probability Theory (E. Lukacs)
208
Martingales ~ Valeurs Vectorielles Application ~ la D~rivation (M. Metivier)
239
Atomes Conditionmels d'Espaces de Probalite de l'Informarion (J. Neveu) 256 On Markov Processes whose Shift Transformation is quasimixing (F. Papangelou) 272 Remarks on the Poisson Process (A. R@nyi)
280
Sums of Markov-chains on Finite Semigroups (L. Schmetterer) 287 On Superefficiency (L. Schmetterer) The Explicit Martin Boundary Constructions
291 (F. Spitzer) 296
Structure des Lois Indefiniment Divisibles ~ £ ~ = dans un Espace Vectoriel Topologique (separe) X (A. Tortrat) On a Paper of J. G. Sinai on Dynamical Systems (H. Zieschang)
S(~I) 299
329
-
I
-
Remarques sur les Theoremes de Bochner et P.Lev~ par A.Badrikiam
Dams le calcul des probabilit~s fonction charact~ristique.
"classique" un grand r~le est jou6 par la notion
Et dams cette direction,
les th~or~mes
de
de Bochner et P.L&~y
sont constamment utilis~s. Nous nous proposons ici de g6n~raliser ces r~sultats pour les variables al~atoires prenant leurs valeurs dams un espace vectoriel topologique
localement convexe ( en
abr6g~ e.l.c.) Tous l@s espaces vectoriels topologiques convexes,
que l'on consid~rera donc seront localement
s~par~s et auront pour corps ~e base R.
On ne consid&rera,
pour simplifie~,
que de probabilit&s,
pu tout aussi bien ~tudier les mesures positives
sur ces e.l.c, mais on aurait
bounces.
I - Notions Prem~m~nalres I - Probabilit~s
compactes sur les espaces compl~tement
r&guliers
Le Cam ou Varadarajam
Boit X un espaces topologique lera probabilit@ la propri~t~
compacte ( ou de Radon,
r@gulier, ~
sa tribu bor@lienne.
ou tendue ) sur X une probabilit@
On appelsur ~ ayant
suivante :
"quel que soit ~ Une f a i l l e
compl@tement
O, il exi,~eun compact K~ de X tel que P(K E) ~ I -
(Pi) de probabilit~s bor~lienmes
sur X est dire ~ u i - t e n d u e
satisfait ~ la condition suivante: "q~el que soit ~ ~ O, il existe un compact K~ de X tel que: Pi(K~) > ~ - g
pour tout i
si elle
-
On appellera
~
(X)
2
-
( ou pus simplement ~ ) l'ensemble des probabilit&s compactes
sur Xe L'utilit~ de cette condition "~qui-tendue" se justifie par les remarques suivantes: Soit ~
(X) l'espace vectoriel des fonctions continues, ~ valeurs r&elles, born&es sur
X, muni de la norme uniforme.
sur
~(X)
sera muni de la topologie de convergence simple
~x).
Alors Le Cam a d~montr~ qu'une partie
~' de
~(X)
&qui-tendue est relativement
compacte. Ce r@sultat admet,
dams certains cas, une r@ciproque, par exemplez
Si X est un espace m&trisable complet une pattie
~' de ~
est relativement compacte
si et seulement si elle est &qui-tendue ( Varadarajan ). Tous ces r@sultats seront appliques au cas o~ X est un e.l.c., disons E. Toutefois il est rare qu'on air ~ faire directement ~ des mesures bor~liennes sur
E
mais seulement ~ des "distributions faibles" ou "mesures cylindriques" que l'on d@finit de la mami@re suivante: Consid~rons F Iet
la base de filtre sur E form&e des sous-espaces de codimension finie. Si
F 2 sont deux tels espaces et si F I c
F 2 , on d&signe p a r ~ F 2 F 1
l'application
canonique de E/F 1 sur E/F 2 . Supposons que pour tout sous-espace F de codimension finie, on se donne ume probabilit& sur l'espace E/F , soit PF ; la famille (PF) satisfaisant ~ la condition de compatibilit~
F1 c F2 ~
PF2
= V F2F1 (p FI)
on a alors un syst~me projectif de probabilit&s qu'on appelle distribution
faible
( ou mesure cylindrique ). U n tel syst~me projectif de probabilit&s simplement additive sur l'alg~bre
~
d~finit alors une fonction d'ensembles
des ensembles cylindriques de E d&finie par:
"Une partie A de E appartient ~ ~
s'il existe un sous-espace de codimension F
et u n ensemble bor@lien A F dans E/F tel que
A
="n-F1 ( A F ) "
( od ~rF d&signe l'application canonique de E sur E/F )
-3-
B'il existe une probabilit& compacte P sur E telle que PF =
~TF(P) pour tout F on dit
que le syst@me projectif des PF d~finit une probabilit~ compacte. Cela ~tant, on a l e r~sultat fondamental suivant: Soit (P~) une famille de distributions faibles sur E , pour que les (P~) d~finissent une f~m~lle (pi) ~qui-tendue de probabilit~s bor&liennes sur F , il faut et il suffit que la condition suivante soit r~alis~e: ( C ) : Pour tout
~ ~ 0 , il existe u n compact K E ~
E tel que pour tout F de
co-
dimension finie et tout i
PFi (~F Cette conditiom ( C ) p o u r la premi@re
(K~)) > 1 - E
a 6t~, ~ notre connaissance,
introduite dams la litt~rature
fois par E.Mourler.
2 - D@finitions relatives aux e.l.c. a) Diff@rents es~aces norm@s associ@s un e.l.c. Solt d'abord V u n voisinage de z@ro convexe @quilibr@ ouvert et PV sa jauge "fonctionnelle de Ninkowski"
). Soit enfin ~
E form@ des x tels que Pv(X) = 0 . ~ complet ) male son compl@t@
le quotient de E par le sous-espace de
n'est pas complet en g@n@ral ( re@me si E
sera not@ par ~
Le voisinage V sera dit hilbertien
( ou
si ~
Soit maintenant A rome ~artie de E , born@e
est
.
est un espace d'Hilbert. convexe @quilibr@e et ferm@e. Soit E A
l'espace vectoriel engendr@ par A , muni de la norme jauge de A PA(X) : inf Si A est complet ( d o n c
{ ~
, ~
g R+ , x C
~A
en particulier si A est compact ) E A est complet. En parti-
culier si E est le dual d'un e.l.c, et si A est une pattie de E convexe ~quilibr~e faiblement ferrule et ~quicontinue E A est un Banach. A sera dire hilbertienne si E A est un hilbert, Eemarquons
enfin que le dual de ~
( V voisinage s'identifie g ~ o
-4!
od E
!
d6signe le dual de E , V'le polaire de V )
b) 0p6rateurs d'Hilbert-Schmidt Soient H 1 et H 2 deux espace d'Hilbert et A une application lin6aire continue de H 1 dams H 2 . A est dite d'Hilbert-Scbm~dt que la famille
IIA (e~)II
s'il existe une base orthonormale ( e ~ )
telle
2 soit sommable:
Z II A e e<~ ) II 2
<
00
Comme la quantit6 qui figure dams le let membre ne d6pend pas de la base orthonorm6e choisie dams H 1 , la notion d' op6rateurs d'Hilbert-Schm4d ne d6pend pas de la base choisie.
On pose,
II .4.112: c Z IIA (eo(~11 2 ) ~i2
Soit maintenant Hl et H 2 deux espaces pr6hilbertiens
et A une application lin6aire
continue de H 1 dans H 2 . Soit H~ etH~2 los compl~t6s de H 1 st H 2 respectivement et le prolongement de A ~ H1 i A sera dit d'Hilbert-Schm~dt si A l'est e~ l'on posera:
IIAll2
:
II~ II2
En consid6rant les couples en dualit~ (H1,H I) et (H2,H 2) A est d'Hilbert-Scbm~dt et seulement si sa tramspos6e ~ : H 2 ~
II A II 2 =
(Ai) du pr6hilbertien H 1 darts le pr6hilbertien H 2 est dit
( en abr6g6 6qui H.S. ) s'il existe M > 0
llAill2
et l'on a :
IIA'II2
Ume famille d'applicatioms 6qui Hilbert-Schmid
51 est d'Hilbert-Schmidt
~
M
tel que
pour tout i
Soit maintenant E un e.l.c, et H 2 un espace pr6hilbertien et soit A u m e lin6aire continue de E darts H 2 . A est dire d'Hilbert-Schmidt print6 suivante
si
application
si elle poss@de la pro-
:
"il existe un pr6hilbertien H 1 et une application d'Hilbert-Schmidt Q( de H 1 dans H 2 tels que la factorisation ci-dessous air lieu
-
5
-
HI
( ~
~
H2
@tamt continue )
Une fAmille d'application lin~aires (Ai) de E dams H 2 est dite ~qui H.S. s'il existe un pr~hilbertien H 1 et une f~m~lle @qui H.S. d'applica~ions de H 1 dams H 2 telles que
E
~
~
HI
~
~ H2
L'e.l.c. E est dit nucl~aire s'il poss~de les propri~t~s suivantes : -
E poss@de un syst@me fondamental de voisinages de z@ro pr@hilbertiens
- Pour tout voisinage de z6ro hilbertien U il en existe un autre V tel que V C
U
et que l'applicationE V -~ E U soit d'Hilbert-Schm~dt. La deuxi@me condition @quivaut ~ dire que toute application lin~aire continue de E dams un espace pr~hilbertien est d'Hilbert-Schmid.
$ - La fonctiommelle carat~ristique ( ou transform~e de Fourier )
L'un des principaux moyens de se donner une "distribution faible" sur un e.l.c. E est le suivamt : on se donne une application
~ de F ( dual de E ) d a n s
C
ayant les pro-
pri~t@s suivantes : a)
~(0)
= 1
et
~ est de type positif
b) la restriction de ~
~ chaque sous espace de F de dimension finie est continue.
C'est alors ume consequence immediate du theorem de Bochner sur ~n que la donn~e d'une telle ~ d~finite une distribution faible unique sur E. R~ciproquement, route distribution faible d~finit une telle fonction ~(y)
=
~
9~ de F darts ~ par la formule
e is Py(dS) = ~ E
ei~X'Y~P (dx)
-6-
OdPy
d~signe la probabilit@ sur le sous-espace de E quotient de E par le sous-espace
orthogonal ~ y ( qu'on identifie ~ R ). I1 existe domc une bijectiom entre les fonctioms complexes sur F satisfaisant ~ a) et b) et les mesures cylindriques sur E. L'om peut alors se poser le probl~me suivant
:
Soit ( ~ i ) ume f~m~lle d'applicatioms de F dams ~ b)
satisfaisant aux c~mditions a) et
, et (Pi) les distributioms faibles correspondantes.
Trouverume
condition topolo-
gique sur l e s ( ~ i ) @quivalemte au fait que les (Pi) d~finissent une famille ~quitendue de probabilites sur E . Une r~ponse partielle sera domm6e au mum§ro suivant.
II - Th~or~mes de Bochner et P.Lev~
Soit E un e . l . c .
- F
son
dual
O n supposera dams tout ce qui suit que E satisfait ~ la comditiom suivante: (H)
E poss~de un syst~me fondamental de parties compactes hilbertiennes
partie compacte est contenue dams une partie compacte hilbertiemme
( i.e. route
)
Proposition I Soit E un e.l.c, satisfaisant ~ la condition (H) Soit F c son dual muni de la topologie de convergence umiforme sur los pattie compactes convexes de E . Soit (Pi) ume famille @qui-tendue de probabilit6s sur E , ~ i leur fonctionnelle caract~ristique. Pour tout & ~
0 , il existe u n Hilbert H E , ume famille
6qui H.S. d'op~rateurs de F dams H E et u n mombre
( is symbols
II' II d ~ s i g n e l a morme dams H £ )
V ~
0 tel que
-7-
D~monstration Soit K~ une partie compacte hilbertienne de E telle que Pi(K~) Posons H~ = E K E
- K~
> 1 -
~ pour tout i .
est alors un voisinage de z@ro hilbertien dams F . Consid@rons
la famille B i d'applications lin@aires de F dams E K~ /K~
y-~
<x,y,>
x
sl y~et Y2 ont ~me i,,,~gesO.~e F ~
Pi (dx) = B~i (y)
B~ 0"I) -- B ~ (Y2)
Donc B Ei d&f~wSt une application B i de F K; dams E K~ Cette application est &videmment continue car si ~ ~ F K# et s i y
d6signe un repr~sen-
taut ~e
s%
Elle se prolonge doric de mani~re unique en une application lin~aire continue de F K~ darts H E • Identifions maintemamt F K~o et H~ (Ke ~tant une partie compacte hilber tienne ). On en d@duit une application lin~aire de H~ darts H~ ; cette application n'est autre que /K~
~'~
(od le symbole (. I •) d~signe le produit scalaire dams H 6 ) Bi
est ~videmment hermitienne de type positif. Soit ~
Bt " Je dis que la f~m~lle (~E)
sst ~qui H . S . .
la racine carr6e positive de
Bolt en effet (Xj)l<" ~ n une famille
orthonormale finie a~ns H e
n
A,£
n
/K
--
, l( J'
=
( ~
I (xsl x) 12 Pi (~) P2Ke(X) Pi (dx)
<.
Pi( K~ )
D' od le r6sultat Posons maintenant
Ai
=
o ~K~
F sur F K£o et F K~ est identifi@
~ un
(~-K~ est l'application canonique de sous-espace de H E )
-8 Les A~ sont par construction ~qui H . S . . trouver V
-
Ii reste ~ d~montrer maintenant que l'on peut
tel que
IIAi
cy>]l 2
< ,7
>
I ~1 - ~'i C~O]~
E
P o u r cela , on op@rera comme suit : ~2~ Soit
~ +
tel que
It I < 6 ~ 1 1
- e it I <
~
et posons
~=
T
pour tout i } et posons
E6
=
( x;
x E K£
;
Ii est alors clair que
yE v(~)<
~
x,y>2 Pi Cdx) ~
O n en conclut tout d,abord que
$2
Pi C K~ -E~ )WV = ~
D 'od
PICK~-E~) < C ela 6tant si y E
~
pourto~i
V (~) , l'on a
11 - ~iCY) l.C/E lei<x'y~ - l I P i Cdx)~ 2P i C E - K E) + 2 P i C KE -E~ ) ÷ ~-
Pi C%~
On a donc d6montr~ que £
I Ai Cy)I1~?
i,
-
c:~l< E
C.Q.F.D. Nous verrons que la proposition (I) admet une r~ciproque importamte. Avant de la d~montrer,
raisons une remarque
:
Soit F I um esoace vectoriel de dimension finie et soit F 2 un sous-espace de F 1 munl d'une structure hilbertienne dont le produit scalaire sera not~ (. I .) • projecteur de F 1 sur F 2 .
Soit~um
-9-
Si Yl ~
F1 on p o s e r a y 2 = ¢ r ( Y l )
La form~le
(Y~, Y~) -" (Y21Y~) definit une forme bilin6aire sur Fq,positive sym~trique (et d~gSn6r6e si F 2 # Fq ) que l'on d@signera encore par le symbole (. I .) • Soit enfin A 2 un op@rateur lin6aire de F 2 darts F2, hermitien de type positif , c'est-~dire ( A2Y 2
1
Y2 )
=
('Y2 I A2Y2 ) st ( AlY21 y 2 ) ~
0
Vy2, y~ a F 2
L'op@rateur A 1 de F I darts F I ( et m~me de F 1 dans F 2 ) d~fini par A 1 = ~ o ~ ales
2
,
propri~t~s suivantess #
( ~Yl
I Yl )
=
( Y 1 I AlYl ) st ( ~ Y l '
Yl ) ~
0
Soit E 1 le dual de F 1 , E 2 celui de F 2 E 2 sera muni de la structure hilbertienne d6duit de celle de F 2 et 'lon d~signera parle m~me symbole les produits scalaires sur F 2 et E 2 E 2 sera identifi@ g u n
sous espace de E I
Cela 6rant, soit ~ u n e
fonction d~finie s u r f I ~ valeurs d a n s ~ ,
continue et de type
positif et telle que ~ (0) = I .~est alors la fonctionnelle caract@ristique d'une probabilit@ (bor@lienne) sur El, soit PI " Ces notations ~tant ainsi fix~es, on peut 6noncer la r~sultat suivant.
Lemme I
si
( d'apres" Prokhorof )
( A l y ~ I Y~ )-< 1
). I1 - ~ ( Y ) I < ~
(~)
Pour tout sous ensemble bor~lien B 1 de E 1 ext@rieur
u = {-'~
--~F 2
Ilxll
< c] ,E
+
1'on
2
a:
-
10-
D~monstration 11 suffit de d ~ m o n t r e ~ l e
r~sultat pour B 1 compact.
L ' o n va tout d'abord construire ls structure hilbertienne C ne rencontre P o u r cela,
doric B 1 compact.
sur E 1 une structure hilbertienne
initiale
et pour laquelle
;
E Iest
:
alors somme directe
( alg@brique
) de E 3 et E 2 .
dont la norme sera notre
Munissons E I de la structure hilbertienne
somme hilbertienne
et E 2 et consid~rons
UI
Alors U =
la famille
=
~g~+
U~
xI E
x 2g
et la famille
il existe
II "
II
"
extreme de celles de E 3
d'ellipsoides
{ xI ;
x I = x2+x 3
B @tant compact, Munissons
sure 2
la boule de centre 0 et de rayon
Munissons E3 d'une structure hilbertienne
;
induisan~
pas B 1 .
on op~re comme suit
Soit E 3 = F 2
Supposons
E1 ,
C2
E 2 ; x 3g
E3 )
+
12
(~) a la propri~t~
~ o tel que U ~ o ~
d'intersection
fini
B1 =
alors F de la structure hilbertienne
dont la boule de rayon C est U 4o
Cette structure F 1 sera mumi
Cette
hilbertienne
de la structure
structure hilbertienne
ales
propri~t~s
hilbertienne ales
- l'op@rateur~
est l'op~rateur
On designera par ~es s~boles normes
sur E 1 et F I relatifs
C e l a &rant,
il est clair que
P1 (B1) ~ P1 Or la fonction Xl-~ e
(
duale de celle dont vient d'etre muni E 1 •
propri~t~s
- elle induit sur F 2 la structure
•
requises.
suivantes
:
initiale
de projection
I. )' e~
aux structures
orthogonale
II" II' les hilbertiennes
produits
sur F 2
scalaires et les
ainsi introduites.
z
{11=III ~
c)
V'~-I
( I - e
est la transform~e
PI (a-~-)
de Fourier de la probabilit@
-11
-
gaussienne sur F1 s
0~
e
(2,.-)n/2 ( od n e s t
I12 IIo ~III'2
la dimension de F 1 ) et dy I e s t
dy1
la mesure de Haar dormant la masse 1 au
pay6 unit@. Doric
e-I"211 11 '2 P1
Cn
=
(2~r)n/2 od < x I , yl >
d@signe l'accouplement canonique entre El, et F 1 ( pour la structure
hilbertienne ). Echangeant alors les int@grations, on a :
j~E1
e
-c-
dy 1
?(yl ) e
PI (~I) : ~2~)n/2
1
\
Ii r~sulte d'autre part de la condition (1) du lemme que
I I-
?(yl)l<
E
et, en vertu de la construction de
+ 2 (A1 Y l l
Yl )
pour tout Yl ~ 1
(. ~ .)'
(A1 Yl I Yl) : (AlYl ! Yl )' Donc
I1 _lp ( y l )
Iz-- ~"
+ 2 (A 1 Yl I yl ) '
L'in~galit@
")
,.J~E1 (1 - e
:PI ( ~ I )
Cn
f
J'1I1
- ,f'cyl~l
e
-1/2 IIc ylll '2
(2)n/2 donne
~E 1 jl -e
_I,2
/2
II IP1 (2~-) n/2
1
(~ ÷ 2(Ay I I Yl ) ') e
-1/2 lie Yl 1l'2 ¢71
-12Or, d'apr@s l'in@galit@ bien connue
O
/(
AlYl'
-112 ( c y l l
Yl )' e
yl )'
d,V1 : Tr ( AC -1)
(2~-)n/a od C d@sigme um op@rateur hermitien, de type positif et inversible. On d@duit : V~- )
( ~1 +
2 Tr C2 AI
PI (~I) ~ V~--C.Q.F.D.
Remarque,
Ii est clai~ que Tr ~
= Tr ~
Corollaire 1 Sous les hypotheses du lemme 1, s'il existe um op@rateur tel qu'en posan~ U 1 = U 2 o ~
hermitien U 2 de F 2 dams F 2
, l'on air:
II ~, Yl II .< 1 ,,
)-
I1
-
~'(yl
)
I <.
alors P1 (B1) ~
A 2 = U~2 ; alors ( A 1 Yl' Yl ) =
e~
~
A~ =
II~ II~
C.Q.F.D.
Corollaire 2 S 'il existe um Hilbert H ( avec la norme
II ° I lH ) et une applicatio'n
V2 de F2 aRl.,~
H tel que
II v2 o ~ (y~)II -< ' Ve
P(BI) <~
V2-I
)'1 I-
if(y1) I.<
÷ II ,'~o~II ~ )
alors
-
13-
D~monstration En offer V 2 se factorise comme suitz
u2
w2
F2 ,
~
F2
~
H
11~2112 ~ IIv211~ et W 2 est 1t"2 <~)11~ : l l ~ < ~ II ; o = tout ~
O~ U 2 est hermitien positif et tel que
isom6trique, c'est-g-dire
partiellement
<,~
Lemme 2 Soit F u n
e.l.c., E son dual. Soit ~ i
une famille de fonctions d@finies sur F, qui
sont les fonctionnelles caract6ristiques demesumes cylindriques s u r E . Supposons qu'il existe une famille (A i) d'applications lin@aires de F darts un Hilbert, de type 6qui H.S. se factorisant comme suit
( avec I I A I I ~
-<
M
pour tout i, et U voisinage hilbertien ) et telle que llAi YlIH
~<
1
Alors pour tout ensemble oylindrique C c E in6galit@s
~
) II -
~i(y) l ~< £
tel que C c E ~ (g
+
2
(~
U ° ) on a l e s
)
(Y
iel)
Pi (c).< VT_ - ~ ( U ° d~signe Is polaire de U dams E )
D~monstration Nous ferons la d~monstration en supposant, pour ne pas surcharger les notations, que l'on s'est donne une seule fonctionnelle caract~ristique ~ •
-14-
Soit F 1 un sous espace de F de codimension finie, et soit ~-1 l'application canonique de E s u r e
( F ~ ) . Consid@rons le diagramme A
F1 II existe
~ F
~ FU° ~
H
un sous espace F 2 de F 1 tel que la restriction ~ F 2 de
est injective ( il suffit de prendre un suppl@mentaire de Ker ~1
~1
s F 1 - ~ F UO
)
Soit ~r un projecteur de F 1 sur F 2. Munissons F 2 de la structure hilbertienue induite par FU . En remarquant que l'application F 1 -~ H qui
figure dams le diagramme se
factorise au moyen de ~r par l'application F2-~H : on peut appliquer le corollaire (2) du lemme (I), Pour tout bor@lien B 1 de E 1 = E / F 1 , ne recontrant pas la boule de rayon ~ on a z
~e
PI
( O~
I
(BI) =
V21122 ) ~
V 2 d@signe l'application F 2 - ~ H
de E2,
+ 2
)
Donc, pour tout ensemble cylindrique C ayantune base dams E 1 et contenu dams E / ( ~ U °) ona
!
~-~
M2
P (c) .< - I
.< Le lemme
(Z
+ 2 Z---,2-)
M
(2) est donc compl@tement d@montr@.
Cela @tant, on peut @noncer
( et d@montrer la r@ciproque de la proposition ( 1 ) )
Proposition 2 Soit E un e.l.c., F c son dual muni de la topologie de convergence compacte. Soient ( ~ i ) une famille d'applications de F dams ~ ayant les propri~t~s a) et b) de 1-3 • g Supposons que pour tout £ > 0 , il existe une f~m~lle A i ~qui H.S. d'op@rateurs de F dams un Hilbert H E telle que II Ai Yll H
~<
I
~
I I - ~i(y) I ~ E
-15-
Alors les
~i
sont les fonctionnelles caract~ristique d'une famille @qui-tendue de
probabilitSs bor@liennes s u r E . Demonstration ~+ et soit
Le dual de F i est, d'apr@s le th@or~m de Mackey, l'espace E. Soit 6
~_~ Soit
(A~) une fAmille @qui H.S. d'applications de F darts un Hilbert H V admettant la
factorisation F
11/3i112 ~< M >1~ - Fi(y) l ~< E
( U voisinage de z@ro hilber~ien, et telle que
II A~ y
I1~
Soit ~ ~ R + tel que
~/'~ - 1
~
2
D'apr@s le lemme (2), si O est un ensemble cylindrique de E ext@rieur
¢e Pi (c)<-
V; ~_
¢e-~ I
(
~Ve
~ +
La fsm~lle (Pi) de mesures cylindriques s u r E
2
U 0
, on
a
~-~ ~
) :
E
pour touti
sa~isfait ~ la condition (C) ; elle
d@finit donc une famille @qui-tendue de probabilit@s bor@liennes sur E. C.Q.F.D. Groupant les propositions
(I) et (2), on obtient le th@or@m suivant
:
Th6or@m q Soit E un e.l.c, quasi-complet ayant un syst@me fondamental de parties compactes hilbertiennes
;
F c son dual ~,n~ de la topologie de convergence uniforme sur les
parties compactes convexes de F . Soit (Pi) une famille de distributions faibles sur
|
-16-
E, ( ~ i )
leur fonctionnelle carac~@ristique.
Pour que la famille (Pi) d@finisse une
famille @qui-tendue de probabilit@s bor~liennes sur E, il faut et il suffit que les ~i Pour
remplissent la condition suivante; tout £ >
F darts H
E
E O, il existe un Hilbert H E et famille A i @qui H.S. d'applications de
telle que g II Ai
x II ~ H2
1
pour tout i
) I 1 - ~ i (x)l ~ E
L'on peut maintenant donner une r@ponse plus pr@cise ~ la question pos@e dams I-3Nous aurons besoin pour cela d'une d@finition. D@finition Soit F u n
e.l.c, muni de la topologie ~ . On appelle S-topologie associ@e ~
topologie de Sazamov (et on note
~S)
~ou
la topologie localement convexes la moins fine
sur F pour laquelle les op@rateurs de type Hilbert-Schmidt de F dams les Hilbert sont continue. Soit H u n
Hilbert et soit
sont
~qui-continues, elle sont @qui H.S.
~S
(Ai) une famille d'applications de F darts H. S i l e s
R@ciproquement, une famille finie d'applications de F dams un Hilbert H e s t si elle est
(Ai)
@qui H.S.
~ S @qui-continue. F, est nucl@aire si et seulement s'il y a 6quivalence
entre les notions de famille @qui H.S. et d'ensemble ~ @qui-continue. Avec ces notations, le th@cr@m (1) donne pour corollaire:
Corollaire 1 Soit E un evl.c, ayant un syst@me fondamental de parties compactes hilbertiennes; soit F son dual. Soit ~ u n e
fonction d@finie sur F ayant les propri@t@s a) et b) de I-3 •
La condition n@cessaire et suffisante pour que ~
soit la fonctionnelle caract@ris-
tique d'une probabilit@ bmr@lienne sur E est qu'elle soit uniform@ment continue pour la S-Topologie asscci@e ~ la topologie de convergence compacte.
Cerellaire 2 S o i t E tm e . l . c . ,
F s o n d u a l rim.hi de l a t o p o l o g i e
de l a c k e y ,
~ ; si pour la topologie
-17-
affaiblle E poss@de un syst@me fondamental de parties compactes hilbertiennes, identit6 entre fonctions sur F satisfaisant au x conditions a) et b), continues,
et fonctionnelles caract@ristique
~S
il y a
@qui-
de probabilit@s compactes pour la topo-
logie affaibl~e. C'est ~mm@diat, car
~
est la topologie de convergence uniforme sur les parties con-
vexes compactes pour la topologie affaiblie.
Applications I - Th@or@m
de type
"Minlos"
Si F e est nucl@aire, faible sur E, ~ i @qui-tendue,
l'on a l e
r~sultat suivant
: Soit (Pi) une famille de distribution
leur fonctionnelle caract&ristique. Pour
il faut et il suffit que les ~ i
que la famille Pi soit
soient @qui-continues.
2 - Th@or@m de "Sazono~' Soit E l'espace vectorlel sous jacent ~ un espace d'Hilbert, on le munira de la topologie affaiblie ~
. Le dual de ( E , U )
muni de la topologie de convergence compacte
peut ~tre identifi@ ~ E muni de la structure hilbertienne, Le corollaire
soit ( E, ~ ).
(2) s'applique dams ce cas et l'on obtient le r@sultat:
" Une application
~
de E dams
~
, de type positif
( et telle que ~ ( O )
= I
)
est fonctionnelle caract@ristique d'une probabilit@ tendue pour la topologie affaiblie si et seulement si
~
est continue pour la topologie sur E la moins fine rendant con-
tinues les applications d'Hilbert-Schmidt Comme d'autre part bor@liennes
de F darts F ."
( cours Schwartz ) il est bien connu que sur un Fr@chet les mesures
compactes sont les memes pour toutes les topologies compatibles avec la
dualit@, on retrouvele th@or@m de Sazonov.
Remarques Dams
ce cas d'un Hilbert, il n'existe pas de r~sultats du genre de Minlos pour la
topologie
~ S , c'est-~-dire
sont fonctionnelles
qu'il
existe
une famille
( ~ t ) de f o n c t i o n s
caract~ristiques d'une famille @ q u i - t e ~ u e
sur E qui
(Pt), et telle que
- 18-
les (~i) ne solent pas
~ S _ ~quicontinues. En effet.
Gardons les notations ci-dessus et supposons que E poss~de une base orthonorm@e non d~nombrable (et) t E T" Soit Pt la probabilit~ donnant pour masse 1 au point (et). La famille (P$) est ~qui-tendue
( pour la topologie ~
)
Dn a d'autre part ~t(y) = e i(yJet) Supposons que les ~ t
sont
~S
~quicontinues; il existerait un op~rateur H.S. de E
darts E solt A tel que
Jl A yll < Doric Ay = 0 Soit donc Donc
1
) #1 - ei(ylet) I ~ E ~ y~
Ker A ;
~
~
~ 1 - e i~(Ylet)I~Epour tout
Donc
y # et ) =
Donc
Ker A
On peut d ~ m e
~. I 1 - ~t(y) I ~
=
0
~ ~. ~
y
g
Ker A
et tout t g T
pour tout t, et y = 0
{ 0 ) et A est biunivoque, ce qui est contradictoire.
d~montrer qu'il n'existe aucune topologie ~ '
ayant la propri@t@
suivante : les ~
i sont ~
topo~ogie ~
.
' ~quicontinues si et seulement si les Pi sont ~qui-tendues pour la
-19-
Biblio~raphie
Kolmogoroff
"A n o t e an the p a p e r s of M i n l o s and P r o k h o r o f f " T h e o r y of P r o b a b i l i t y and i~s a p p l i c a t i o n s - 1959 - pp 2 2 1 - 2 2 3 ( t r a d u i t du r u s s e "Convergence
L e C a m L.
)
in d i s t r i b u t i o n of s t o c h a s t i c p r o c e s s e s "
U n i v e r s i t y of C a l i f o r n i a - P u b l i c a t i o n s 1957
in S t a t i s t i c s - V o l . 2 -
Pp 207 - 236
" G e n e r a l i z e d r a n d o m p r o c e s s e s and t h e i r e x t e n s i o n in m e a s u r e "
M i n l o s R.A.
S e l e c t e d t r a n s l a t i o n s in m a t h e m a t i c a l S t a t i s t i c s a n d n ° 3 - American Math.Society - Providence
"Les & l e m e n t s a l ~ a t o i r e s
M o u r i e r E.
Institut
-
Annales
1953 - P a r i s - p p . 1 6 2 - 2@@
"The m e t h o d of c h a r a c t e r i s t i c f u n c t i o n a l s "
Prokhoroff
( t r a d u i t du russe)
darts u n espace de B a n a c h "
Henri Poincair@
Probability
- P r o c e e d i n g s of the
F o u r t h B e r k l e y S y m p o r i u m on M a t h e m a t i c a l S t a t i s t i c s and P r o b a b i l i t y Vol.
Prokhoroff
et
Sazonov
II - U n i v e r s i t y of C a l i f o r n i a P r e s s - 1961 - p p . 4 0 3 - 419
"Some results associated with Bochnes'l theoreW' T h e o r y of p r o b a b i l i t y and its a p p l i c a t i o n s - V o l VI - 1961 pp. 82 - 86
S a z o n o v V.V.
( t r a d u i t du russe
" R e m a r k s of c h a r a c t e r i s t i c its a p p l i c a t i o n s - V o l I I I -
S c h w a r t z L.
" M e s u r e s de R a d o n Ron~otyp@
V a r a d a r a j a n V.S.
)
fumctionals"
- T h e o r y of p r o b a b i l i t y and
1 9 5 8 - pp. 201 - 205
darts les e s p a c e s t o p o l o g i q u e s "
( t r a d u i t du russe)
- C o u r s 1 9 6 4 - 65
- Paris
"Meri ha Topologischeskikh ProctramtbAkb" ( Mesures
sur les e s p a c e s t o p o l o g i q u e s
) - M a r e S b o r n i k - T o m e 55
( 97 ) : I - pp 35 - 1OO - M o s c o u 1961 V a k h a n i a N.N.
"0 X a p a k t e r i c t i c h e s k i x ~ i n ~ c i o n a l a x " C e n t r a V . 1 - T b i l i s s i - 1965
-
20
-
Recent developments in axiomatic potential theor~ Heinz Bauer
")
In the last ten years M.Brelot's axiomatic theory of harmonic functions together with the contributions of several authors has become very powerful. Unfortunately,
it does
not apply to parabolic differential equations. Therefore the author has tried to modify B~elot's theory in such a way that the application to parabolic differential equations becomes possible. It is the intention of this paper to present some of the recent developments in this more general theory, in particular those where the "parabolic structure" can be seen very clearly. For more details see forthcoming lecture notes ~].
$
I
Harmonic Spaces
Let us repeat first the sxioms of the theory and some of its fundamental notions, Let X be a locally compact space with a countable base and consider a mapping ~
which
associates to each open subset U ~ 0 of X a linear subspace B~ U of the linear space (X) = E(X,R)
of all real-valued continuous functions on X. The elements of ~ U
f
will be called harmonic functions in U. A set V c X is called regular relatively compact with boundary V * ~
if it is open,
0 and if every function f ~ ~(V~
permits a
uniquely determined solution H~ of the Dirichlet problem which is ~ 0 if f is ~ Hence for ea~a point x
HVCx)
~
=
jf
~ V there is a m e a s u r e ~ Vx ~ 0 on V ~ defined by
~V
is called the harmonic measure
for all f
E ~ ( V ~)
associated to V and x.
*) At the conference this paper was presented by M.Sieveking
0.
-
A function u: U ~ ]
-oo, + ooS
21
-
on saopen set U C X is called hyperharmonic
if it
is
lower-continuous and if
Ju d~ ~ u(x) holds for all regular sets V c V c U and all points x g V. The s~mbol ~ U
will de-
note the set of all hyperharmonic functions on U.
The space X is called an harmonic space
( with respect t o t
) if ~
satisfies the
following four A~ioms: i) If) ixi)
U -~o
U is a sheaf
( in the sense of R.Godement )
The regular sets form a base of the space X. For each increasing sequence (hn)n=1,2,.. ° of harmonic functions in an open set U the upper envelope h = sup h n is harmonic in U if it is finite dense subset
iv)
on a
of U.
a) On every relatively compact open set U ~ ~ there exists a strictly positive function h E ~
U •
b) Every pair x,y of different points of X is separated by functions u , ~ in the sense of u(x) v(1) ~ u(y) v(x).
We introduce immediately a stronger form of harmonic spaces. A function s E ~ called superharmonic
is
in U if it is finite on a dense subset of U. The set of all
(resp. of all non-negative ) f u(resp.
U
superharmonic functions on U will be denoted by
+ ~ U ) . Functions p C
on X satisfying 0 ~ h ~ p
+~U
such that h = 0 is the only harmonic function h
are called potentials.
If for each point x e X there exists a potential Px satisfying Px(X) ~ 0 the space X is called strongly harmonic.
Strongly harmonic spaces may be also characterized by the
axioms I to IV if one replaces in IV, ~ t h e
set ~
by
+~X
"
An harmonic space is called elliptic if every point x E X has a base of regular nelghbourhoeds V such that the support S ~ V boundary V ~
of the harmonic measure ~ V equals the x
of V. The harmonic spaces studied by Brelot may be then characterized as
-
22
-
the strongly harmonic spaces which are connected and alliptic. We call such a space a Brelot space .
n+l
If one chooses X = R
to be the set of solutions of the heat equation
and ~ U n
u
=
~ ::~+1
'
one obtains a strongly harmonic space which is not elliptic. General classes
of linear
parabolic differential equations leading in the same way to a strongly harmonic space have been established by S.Guber.
The reader is referred to Guber's contribution to
these Proceedings.
2
Absorbing sets and Harnack inequalities
Let A be a subset of an harmonic space X. A is called absorbin~ if it satisfies one of the following equivalent conditions: (i) (ii)
A = u-l(o) for an hyperharmonic function u ~ 0 on X. A is closed and, for each x ~ A and each regular neighbourhood V of x, the harmonic m e a s u r e # ~
(iii)
A =
~x E X:
is supported by A.
u(x) < + ~ }
There are no compact absorbing sets ~
for an hyperharmonic fuction u on X.
6. Since X is locally connected ~ , X and all
connected components of X are absorbing sets. Hence in a Brelot space, X is the only non-empty absorbing set. This explains why absorbing sets do not appear in Brelot's theory. Yet, absorbing sets become important for many problems in the parabolic case. In the case X = R n+1 and the sheaf of solutions of the heat equation all absorbing sets are given by A for
~ ~ [ - co, + ~ ]
=
Ix
£ R n+l
: xn+ 1 ~ T }
. Here Xn+ 1 denotes the ~ + 1 ) th coordinate of x.
Absorbing sets appear in a natural way in the formulation of Harnack inequalities:
-
23
-
Theorem 1 Let ~
be a positive Radon measure on an harmonic space X and denote by AS2 the smal-
lest absorbing set containing the support Sp ofp .Then for each compact set K contained in the interior of ~
there exists a constant ~ = ~ ( K , p) ~ 0 such that
sup h(K) holgs for all functions
~
h~0
~Jh
d~
harmonic in X.
Important special cases are the following twos
~ has a point support and ~ h a s
the
total space X as support. The latter case together with an appropriated characterization of nuclear spaces leads to
Theorem 2 For every harmonic space X the linear space ~ X
endowed with the topology of uniform
convergences on compact sets is a nuclear F-space. In particular, ~ X
is a Montel
spaces and reflexive.
Corollary 1:
Every locally uniformly bounded subset o f ~ X is equicontinuous and re-
relatively compact.
-
Since theorem 1 leads to locally uniformly bounded sets of po-
sitive harmonic functions, one obtains in particular:
Corollary 2:
LetHbe a positive Radon measure on X and let J~# be the set of all har-
monic functions
h ~ 0 on X satisfying
the interior of ~
fh
d~
~ 1. Then ~
is equicontinuous on
•
For the case of a Brelot space X and for ~
=
~ x (point
mass I at x) this reproves
the known result of G.Mokobodzki that the axiom (3') of Brelot is valid.
§
~
Cauchy-Dirichlet problem
In the case of the heat equation we determined all absorbing sets: the are all of the type Ag with T
g[- ~,
+ oo] . The boundaries of the set A~ with real
~
real are the
-
24
-
"horizontal" hyperplanes in R n+q , hence the characteristic planes for the heat equation on which the boundary values for the Cauchy problem are prescribed.
This re-
mark opens the way to the following Cauchy-Dirichlet problem which is presented in full detail in
~2~.
Consider in a strongly harmonic space X an open set U with boundary U ~ ~ by~
~.
Denote
U the set of all potentials p on X such that the restriction of p to ~ is
strictly positive and continuous. Then the following variation of the now classical Perron-Wiener-Brelot-method
can be considered: Let p be a potential i n ~ a n d
a real-valued function on U ~ . Call p-upper function every function u a ~
let f be satis-
fying thefollowing two conditions: a)
lim inf u(x)
>
f(z)
for all z &
U~;
X---Z
b)
u(X)
>I
-~p(x)
for some a~> 0 and all x a U
Define ~f as the infimum of all p-upper functions and define Hf by the equation Hf = - ~f . As in the classical form of the Perron-Wiener-Brelot-method the p-resolutive functions, i.e. to those functions f on U for which
for which ~
this leads to = Hf holds and
Hf --J~f = Hf is finite on U.
Denote by J< ( U )
the linear space of all functions f e ~ ( U
) with compact support.
Then the following theorem holds:
Theorem All functions f ~ ~ ( U )
are p-resolutive with rspect to all p ~ ~D U. The corres-
ponding generalized solution Hfis independent and defines for each x c
Denote by
z) of the special choice of p in ~ U
U a Radon m e a s u r e p Ux on U" by
~p(U *) the linear space of all functions f ~
majorized by some multiple of p E ~ U "
~ ( U ~) for which Ill is
then the following corollary holds:
~) This has been observed by J.KShn and M.Sieveking
-
25
-
Corollary All functions f ~ Ep(UO are p-resolutive and ~Ux - integrable for all x
~ U. Then the
g e n e r a l i z e d s o l u t i o n i s g i v e n by
El(X) = /f d2 Ux
( x eU )
and hence independent on the special choice of p a ~ U "
Therefore f - ~ H f
is a linear positive mapping of the linear space
=
in the linear subspace of all h ~ ~ U
which
(in absolute value) are bounded on U by
some potential p ~ ~OU. A boundary point z c U* is called regular if lira
U ~ x
=
~z
X--~ Z
holds in the vague topology. For this type of regular points the same characterizations are valid as in the case where U is relatively compact. In this case the whole development is independent of p ~ ~ U
and reduces itself to the ordinary treatment of
the Dirichlet problem by the Perron-Wiener-Brelot-method. The importance of absorbing sets in this context comes from the fact that for the complement U
=
~
of such a set all boundary points turn out to be regular. Hence
one obtains the following~heorem
which contains all known results about the Cauchy
problem in the case of the heat equation:
Theorem Let A be an absorbing set with non-empty boundary A* and with complement U Then for every function f ~
~pot(U ~) the generalized solution Hf is the only
harmonic function in U having the following two properties: a) b)
IHfl
is bounded by some P ~ U "
lim Hf(x) X-~Z
=
=
f(z)
for all z ~ A ~
.
-
$
26
-
Relations to the theor~ of Markov processes
Since the paper ~4S of P.A.Meyer it has been an open problem whether for a given strongly harmonic space X there exists a "good" Markov process which is as closely related to the potential theory of X as Brownian motion is to classical potential theory. Since a strongly harmonic space is in general not the union of an increasing sequence of regular sets Meyer's method gives only a local result. Recently, however, N.Boboc, C.Constantinescu and A.Cornea obtained the following global result
E3J
:
Theorem
Let X be a strictly harmonic space on which the constant function I is hyperharmonic. Then a semigroup (Pt) t ~ 0
of kernels on X exists with the following properties:
(a)
Pt transforms functions f ~ ~
(b)
Lim11Ptf- fJl =
0
(X) into bounded functions Pt f ~ ~(X)
(f~CX))
t~o (where
If lJ denotes the usual sup-norm).
(c)
x -~J
Pt 1(x) dt
is a bounded continuous potential.
(~)
The excessive functions of (Pt) coincide with the hyperharmonic functions OonX.
The semigroup (Pt) may then be interpreted as the semigroup of transitions of a Hunt process with continuous paths. This opens the possibility to give probabilistic interpretations of potential theoretic notions.
§
~
Final remarks
It is still an open problem to characterize those Markov processes which are associated to a strongly harmonic space. More interesting and probably also more promessing is the following problem: Construct a more general axiomatic potential theory which englobes more
( if not all )
of Hunt's theory. Such a theory
ksheaf'of cones of "hyperharmonic" functions;
should start with a
the use of regular sets should be avoided.
-
27
-
One could then hope to have applications to non-linear partial differential equations of second order.
References (1)
H.Bauer.
Harmonische Riume und ihre Potentialtheorie. Lecture Notes in Mathematics, Springer-Verlag
(2)
H.Bauer.
( 1966 )
Zum Cauchyschenund Dirichletschen Problem bei elliptischen und parabolischen Differentialgleichungen. Math. ~nnalen 164 (1966),
(3)
N.Boboc,
142 - 153
C.Constantinescu, A.Cornea, Semigroups of transitions on harmonic spaces. Revue Roum.Math. Pures Appl.
(4)
P.A.Meyer
(to appear)
Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory. Ann.Inst. Fourier 13/2
(1963)
357 - 372
-
0ber wesentlich
28
-
indefinite
Spiele
D. Bierlein
FUr Zweipersonen-Nullsummen-Spiele timaler
Strategie
einem definiten punkt:
weicht
schleehtert
entwickelt
wurde die Konzeption
unter der Voraussetzung
Spiel bildet
d.h. das Intervall
Nach dem yon Neumann'schen"Hauptsatz"
her ist die diskrete
Erweiterung
Erweiterung
zu diesen
Spielen,
behoben werden kann,
und Y wesentlich indefinit
mit endlichen
indefinite
der Spieltheorie Strategie-Mengen
yon
so ver-
Spiel ein Spie-
(X,Y,a)
definit,
dem Gegner kampflos
ist die gemischte
preis.
Erweiterung
X und Y stets definit.
Allgemei-
falls X oder Y endlich ist.
bei denen eine Indefimitheit
durch eine gemischte
gibt es bereits unter den Spielen mit abz~hlbaren
Spiele,
ist. Eine Untersuchung
aus,
so gibt er das gauze Indefinitheitsintervall,
zwischen unterem und oberen Spielwert,
(X,Y,a)
In
einen Gleichgewichts-
Legt sich dagegen in einem indefiniten fest,
als op-
des Spieles.
einer der beiden Spieler auf eine Nicht-Hinimax-Strategie
ler auf eine Minimax-Strategie
Im Gegensatz
der Definitheit
jedes Paar yon Minimax-Strategien
er seine Position.
eines Spieles
der Minimax-Strategie
d.h.
solche,
der wesentlich
bei denen
jede gemischte
indefiniten
X
Erweiterumg
Spiele interessiert
beson-
ders aus zwei GrGnden: I. im Hinblick
auf die Frage nach einem vernGnftigen
(oder optimalen)
Verhalten
in solehen Spielen, 2. unter dem Gesichtspunkt In dieser Umtersuchung
indefiniten
Spiele umfa~t.
das sich als idealisierte l~utert, h~herer
durch Komposition Stufe zu gelangen.
definiert,
des Uberholspiels,
beiden Aspekten,
Spiels 0berholspiele
yon Definitheitskriterien.
Spielen und als (uneigentlichen)
Eigenschaften
hang mit den oben erw~hnten Frage, welche
des Arsenals
wird der Typ des "Uberholspiels"
Klasse yon wesentlichen definiten
der Ausweitung
sind,
"Satellitenjagd"
alle
im Zusammen-
2 behandelt;
auf die
3 ein. An Hand eines Beispiels,
interpretieren
von Uberholspielen
Grenzfall
insbesondere
werden in Abschnitt
geht Abschnitt
der eine wichtige
l~Ht, wird das Prinzip
zu wesentlich
indefiniten
Spielen
er-
-
29
-
1. Be~riffe und Bezeichnun~en ( U,V,a )
bezeichne die Normalform eines Zweipersonen-Nullsllmmen-Spiels, bei dem die
Spieler I u n d
2 einen l~in~t ("Strategie") u bzw. v
aus den Mengen U bzw. V w~hlen
und das Ergebnis einer Partie dutch die Zahlung des Betrages a(u,v) von Spieler 2 am Spieler 1 ausgedrGckt ist. In U und V seien strategisch ~quivalemte Punkte bereits identisch; d.h. a(u 1,v) =
a(u2,v) fGr a l l e v ~ V impliZiere
=
u I = u2, analogf~r die Strategiendes Spielers 2.
Der untere Spielwert desxSpieles (U,V) ist a~ (U,V): = s u~ i ~
a(u,v) ,
der obere a ~ (U,V): = v6vinfSuU~ a(u,v) Eine Minimaxstrategie u ~
des Spielers 1 garantiert ihm einen Gewinn a(u~,v) ~ a (U,V)
eime ~ - gute Strategie u~ einen Gewinn a(u~,v)~
a~(U,V)
-~.
Der obere SpielweFt hat die analoge Bedeutung fGr Spieler 2. Ein Spiel ~ falls a~( P )
=
a*(~)
Das Intervall I( F ) : =
( =:W(F)),
und indefinit, falls a~( P ) ~
heist definit
a~(~)
.
~a~( ~ ), a*( F ) ~ nennen wit Indefinitheitsintervall. Im Fall
eines definiten Spieles enth~lt I( F ) nur einen Punkt, den Spielwert W( F ). Eine gemischte Erweiterung eines 8pieles mit Gberabz~hlbar vielen ("reinen") Strategien ist auf verschiedene Weise und mit unterschiedlicher Extension m8glich. Wit definieren allgemein:(P,Q,a) ist eine gemischte Erweiterung von (X,Y,a), falls gilt: (1)
X c P c p: WMa~ mit Tr~ger X 1 , analog f~r Q ~ .
(2)
a(p,q): = ~ a ( x , y )
8p dq existiert fGr alle p ~ P, q ~ Q.
Im Spezialfall der diskreten gemischten Erweiterung
~ D = ( pD,QD,a ) sind alle WMaBe
mit abz~hlbaren Tr~gern als gemischte Strategien zugelassen. Wit nennen ein Spiel ~
wesentlich indefinit, wenn jede gemischte Erweiterung bzw.
jede ~imfassendere gemischte Erweiterung von F
indefinit ist.
Zur AbkGrzung werden wir mit X m ~ X bezeichnen; analo~ ist Y m ~ Y
eine aufsteigende Folge X lc X2c °.. mit ~-~Xm = X m definiert. FGr eine spezielle gemischte Erweiterung
(P,Q,a), bei der X m p-meBbar ist fGr jedes p G P
und Ym q-meBbar fGr jedes q ~ Q ,
w~r 1 ) "X d P" bezieht sich auf die Identifizierung v o ~ W~aB Px ~ P
setzen
~ X mit dem auf x konzentrierten
-
30
-
2. Definition umd Ei~enschaften eines Uberholspieles Wie schon in der Einleitung erw~nnt, besitzt jedes Spiel (X,Y,a), in dem X oder Y endlich ist, eine definite gemischte Erweiterung, n~mlich (PD,QD,a). F~r abz~hlbare X und Y gibt es bereits wesentlich indefinite Spiels. Das n~chstllegende Beispiel ist: Bsp.1.
X und Y sind Exemplare der Menge der natGrlichen Zahlen, I a (x,y)
1 0
=
fGr x > y fthr x= y fiir x <: y
-1
Sieger ist also, wer die grSBere Zahl nennt. Wie leicht nachzuprtlfen ist, l~Bt sich hier das Indefinitheitsintervall
~-1,1~ durch keine gemischte Erweiterung verkleinern.
Das Charakteristische an diesem Spiel soll in der folgenden Definition
eines "Uber-
holspieles" erfaBt werden: Def.: >
(P,Q,a) heiBt ttberholspiel 0 Mengenfolgen X m ~ X
(Abk.: U.) zum Intervall ~ 1 '
und Y m ~ Y
a~ (Pm,Qn)
>
a~ (Pm'Qn) ~
d2] ' falls zu jedem
existieren derart, dab
d2 - ~
fiir alle m > n
und
~1 + £
fiLr alle n > m.
2)
Interpretation eines Uberholspieles: Ist (P,Q,a) ein U. zuE~l, 4 2 ~
, so existiert zu jedem E ~ 0 und natGrlicher Zahl m
eine Strategie pm ~ Pm derart, dab a (pm, q ) > ~ 2 - E
fGr alle q e Qm-l'
und
eine S t r a t e ~ e qm E Qm derart, dab a (p,qm)<~l
+ g
fGr alle p ~ Pm-1
2) Die implizit an X m gestellte B~dingumg der p.MeBberkeit for jedes p E p ist keine echte Einschr~ukung, da jedes ~aB P I ~ ~ B(~
~ I ~ ' X 2 ' ' ' " }) bssitzt
P
mindestens eine Fortsetzung Pl auf
( IBm, Satz 2B ). Die gemischte Erweiterung mit den
~ortsetzungen yon p~ P bei Adjunktion von X 1,X2,.. hat also dasselbe Indefinitheitverhalten wie die nit P.
-
FtLr 41 ~ d 2
31
-
garantieren somit Pm bzw'qm g -fast den jeweils gGnstigen Eckp~n~t yon
[d 1 , ~ 2 S - vorausgesetzt, der Gegner wird Gberholt. Genau wie in Beispiel 1 kommt es f~r jeden der beiden Spieler darauf an, eine mSglichst groBe nattLrliche Zahl zu w~hlen, hier den Index m. Das Spiel hat also den Charakter eines sportlichen Wettkampfes. Da das Ersinnen und Nominieren einer groBen Zahl mit wachsender Anstrengung verknGpft ist, erscheint es als
der Realit~t angemessen, das Spiel als Zweipersonen-Nichtkonstant-
summen-Spiel zu beschreiben
mit Auszahlungsfunktionen ai, bei denen diese Anstrengung
durch Zusatzterme b I (x) bzW.bR(Y) erfaBt wird:
a1(x,y):
: a(x,y)
- b 1(x)
a2(x,y) : =-a(x,y) - b2(Y) Eigenschaften eines Uberholspieles (2.1)
Ist /~ = (P,Q,a) ein U. zuF6/1 ~ 2 ~
a(~)~I~2~
, so gilt
a~( ~);
d.h. das Indefinitheitsintervall Gberdeckt das Oberholintervall~ 1 , ~ 2 ~ . Der Beweis ergibt sich aus der folgenden Absch~tzung: F~ir jedes & > 0 gilt a~(F)
=
sup a~(PmE,Q) ~ m
(2.2)
sup a~(PmE,Q)
=
-p6
£
sup inf a~q m,Q~)-, o~ 1 +
n
Ist /~= (P,Q,a) ein ~. zu [dq ~ ~ # und P b P ,
m
n
Q'>Q, so ist auch (P',Q',a) ein U.
zu ~ 1 , ~ 2 ~ (eventuell auch zu einem gr6Beren Intervall) Aus (2.1) und ~2.2) folgen unmittelbar die beiden weiteren Eigenschaften. (2.3)
Ist
/~ = (X,Y,a) bzw. (P,Q,a) ein ~. zu [ ~ 1 , d 2 ~
mit d l ~
2' so ist jede
bzw. jede weitergehende gemischte Erweiterung von /~indefinit, d.h. /~ist wesentlich indefinit. (2.4)
Ist /~ein U. zu I( /~ ), so ist /~ minimal indefinit in dem Sinn, dab sich das Indefinitheitsintervall dutch gemischte Erweiterung yon /~nicht mehr verkleinern l~LSt.
Bemerkung: Es liegt nahe, die Verallgemeinerung eines U. zu behandeln, bei der in der Definition an die Stelle der aufsteigenden Folgen X m und Ym aufsteigende Systeme X t u n d
Yt m i t t
aus einer al~gemeineren linear geordneten Indexmenge T treten. FGr solche Spiele gelten
-
32
-
aber die Aussagen (2.1), (2.3) und(2.4) nicht mehr allgemein, und zwar gilt in der Absch~tzung im Beweis zu (2.1) anstelle der beidem Gleichheitszeichen nur noch die Absch~tzung ,,_>
H
3. Die Klasse derUberholspiele Unm4ttelbar aus der Definition ergibt sich: 3.1)
Jedes definite Spiel
( mit Spielwert W ) ist (uneigentliches) U.zu
W
.
Ein Spiel mit definiter gemischter Erweiterung braucht aber selbst kein U. zu sein; es gilt vielmehr allgemein 3.2)
Ist (X,Y,a) indefinit und sind X und Y endlich, so ist (X,Y,a) kein U.
Der Beweis folgt mit (2.1),(2.2) und (3.1). Zu den Uberholspielen gehSren nicht nur die - definiten - diskreten gemischten Erweiterungen yon Spielen, bei denen der eine der beiden Soieler nut endlichviele
reine
•Strategien zur VerfGgung hat, sondern auch die diskreten gemischten Erweiterungen der Spiele mit abz~hlbar vielen reinen Strategien: 3.3)
Sind X und Y abz~hlbar,
so ist die diskrete gemischte Erweiterung
pD
ein U.
zu I ( ~ D). Allgemeiner gehSren zu den Uberholspielen zum eigenen Indefinitheitsintervall Spiele,
alle
die slch aus definiten Komponenten aufbauen lassen im Sinn der folgenden
Definitionz Def. s
(P,Q,a) h e ~ t
definit komponierbar, falls Mengenfolgen X m J X
und Y m / Y
ex-
istieren derart, ds3 (Pm,Q,a) und (P,Qm,a) fGr Jedes m definit ist. Eine Abschw~chung der in dieser Definition
stehenden Bedingung liefert schlie~lich
genau die Klasse der Uberholspiele zum Indefinitheitsintervall: Def.: und Ym ~
(P,Q,a) heist fastdefinit komponierbar, falls zu jedem ~ > 0 ~engenfolgen X m ~ I X existieren mit llm
a~(Pm,Q) >
lim a(Pm,Q) - £ ,
m~oO
m ~
lira a~(P,Q~)~ > m~
lira a (p,Qm) m
-
(3,4)
~ist
genau damn ein U. zu I ( F ) ,
33-
wenn es fast definit komponierbar ist.
Der Beweis hierfGr erfordert etwas mehr MGhe. Wegen (2.4) sind also fastdefinit komponierbare Spiele bereits minimal indefinit. Die Vermutung nun, dab eine fastdefinit komponierbare gemischte Erweiterung dann, wenn X oder Y abz~hlbar ist, stets existiertj ist falsch, wie ein Gegenbeispiel
zeigt, das in Abschnitt 4 behandelt wird.
4. Wesentlich indefinite Spiele h~herer Stufe Wird der Prozess ~er Komposition yon Strategienmengen iteriert, indem zun~chst im zweiten Schritt definit oder fastdefinit komponierbare K o m p o n e n t e n X m U n d Y
m als Bau -
steine zur Bildung von X bzw.Y verwendet werden, so gelangt man zu echten Verallgemeinerungen des fastdefinit komponierbaren Spieles Indefinitheitsintervall.
und damit des Uberholspieles
zum
Wir geben ein Beispiel dafGr, dab folgende Situation ein-
terten kann: XmTX,
Yt~Yt/~Y,
(P,Qtm,a) umd ( P m , ~ , a ) folglich ist • t
sind definit mit Spielwerten 1 bzw 0 fGr jedes t u n d
= (P'Qt 'a) definit komponierbar und U. zu
F0,1~ fGr jedes t,
= (P,Q,a) aber ist nicht definit (auch nicht fastdefinit) komponierbar, wesentlich indefinites Spiel 2.Stufe. Dabei ist I ( / ~ )
m,
sondern
= I( /~t ) = ~b , 17 .
Der formalen Definition des angek~digten Beispiels sei eine Interpretation vorausge schickt: Spieler 1 hat die Aufgabe, einen Satelliten abzufangen, gestartet wird. Er kann zu einem Z e i t p ~ t HShe h reicht und den
der yon Spieler 2
t eine Sperre errichten, die bis in die
Satellitemau~er Gefecht setzt, falls dieser im Z e i t p ~ t
der Luft ist und nicht hSher als h fliegt.
Spieler 2 w~hlt die Startzeit ty und fGr
jedes t ~ ty eine endliche FlughShe y(t). FGr t u n d
fGr die HShen sind nur natGrliche
Zahlen zugelassen. X ist also abz~hlbar. Da vor dem Start der Satellit ist, setzen wir der einheitlichen Beschreibung wegen y(t) = ~ richtung einer Sperre im Z e i t ~ n ~ t der ~enge Ytoho
t in
auBer Gef~bw
fGr t < ty. Die Er-
t o bis in die HShe h o entspricht somit der Auswahl
aller Flugbahnen mit Y(to) ~
ho .
Liegt die von Spieler 2 bestimmte
Flugbahn in der von Spieler 1 ausgew~hlten Menge Yth' so gilt der Satellit als unsch~dlich gemacht : Auszahlung a = 1, andernfalls a = 0 . Die Strategienm~ngen und die Auszahlungs~mWtion
ergeben sich also wie folgt:
-
x = {Yth'
t, h
34
-
N j,
Y = {ylN : ( g t y E N
: y(t) = ®
fur t < t y ,
y(t>E
furt
ty
(also far y ~ dabei ist Ytoho : = l y
~ Y
Y(to)
~
x ),
ho~
Es gilt dsnn (0)
Xm:= {Yth: t E N Ytm
(1)
Yt
,
TM
h.~m I /
X,
= ~ y£ Y :
y(t)
oo}
=~y
g Y:
tyg
t~t/Y
(X,Ytm,a) ist definit mit W = 1 fGr jedes m und t (Xm,Yt,a) ist definit mit W = 0
(2)
(X,Yt,a) ist also definit komponierbar, semit U. zu ~0,1~ und wesentlich indefinit
mit dem Indefinitheitsintervall FO,1
Die Ergebnisse in (I) und (2) bleiben beim Ubergang zu jeder echten gemischten Erweiterung erhalten. (3)
~=
(X,Y,a) und jede gemischte Erweiterung davon ist somit komponierbar aus
definit komponierbaren Komponenten. 0hne besondere Mtthe ist zu erkennen, dab /~ wesentlich indefinit ist mit I ( / ~ ) Erweiterung yon ~
( der Grenzfall P
= E0,1~ . Den Nachweis, dab keine gemischte eimgeschlossen )
sich fastdefinit kom-
ponieren l~St, k~nn man fGhren, indem man ein ~ > 0 angibt derart, dab fur jede aufsteigende Folge T1,T2,... von Teilmengen yon Y, die die Bedingung a~(P,Tm) > > 1 - ~ erfGllen (diese Bedingung folgt aus der 2. Ungleichung in der Definition der fastdefinit komponierbaren S p i e l e ) , ~ ' T m ~ Y gilt. FUr 6 ~ 1/3 ist m dieser Nachweis durch ~Konstruktion einer Bahn y ~ Y - Z ' T m gelungen. m Ein wesentlich indefinites Spiel dieses Types liBt sich nicht mehr dadurch charakterisieren, dab es fGr jeden der beiden Spieler darauf ankomme, einen geeignet zu
bestim-
menden Index m6glichst groB zu w/hlen: In Bsp. 2 muh Spieler 1 danach trachten, zugleich t ~ ty und h 9 y(t) zu erreichen, also t und h mSglichst groh zu w/hlen. Verwendet er die unter den angegebenen Umst~nden gr~BtmSgliche Zahl n, um die Strategie Yn,f(n),mit vorher gew/hlter (monotoner) Funktion f(n) zum
Einsatz zu bringen, so
kSnnte Spieler 2 bequem mit der Bahn y(t): = f(t) + 1 stets erfolgreich bleiben, wie
-
35
-
gro2 auch Spieler I n w~hlt. /~hnllch ist die Schwierigkeit fGr Spieler 2.
Ausgehend yon den fastdefinit
komponierbaren indefiniten Soielen als den wesentlich
indefiniten Spielen 1.St~ufe, l ~ t
sich eine Hierarchle wesentlich indefiniter Spiele
aufbauen, wobei die (k + 1)re Strafe yon den nicht z u r k - t e n Stufe geh5renden, aus Spielen der k-ten Stufe komponierbaren wesentlich indefinlten Spielen gebildet wird. Es besteht Grund zur An~,hme, da~ die Stufenleiter nicht bei einer endlichen Stufe abbricht.
Literaturs
[B]
Bierlein
'q~ber die Fortsetzung yon Wahrscheinlichkeitefeldern" Z.WAh~sch. 1
( 1962 )
-
36
-
LA TOPOLOGIE FINE EN THEORIE DU POTENTIEL par Marcel BRELOT
Introduction
1.
-
Je me propose seulement de donner quelques notions de base d~n~ un cadre
axiomatique tr~s g@n@ral, puis divers exemples d'applications en th@orie du potentiel (classique ou axiomatique) avec quelques r@sultats in@dits, mais sans interpr@tation probabiliste. Historiquement (1), la comparaison des notions de point r@gulier pour un ensemble (surtout pour les points-frontiAre au sens du probl~me de Dirichlet) et de point stable d'un compact pour un probl&me de Dirichlet analogue, m'avait conduit A la notion plus g6n@rale d'effilement en th@orie classique (1940). CARTAN ramarqua aussitSt que les compl~mentaires des ensembles voisinages de
x0
e ~ x 0 , effil~s en
x 0 , forment les
darts une topologie qui est la moins fine pour laquelle les po-
tentiels ou fonctions surharmoniques (m@me consid@r@s localement) deviennent continus. I1 l'appelle topologie fine, et les notions topologiques correspondantes sont devenues des outils usuels. Cela fut @tendu ~ des axiomatiques modernes, et une grande partie des id@es de base est valable dams des conditions encore plus g@n@rales que j'expliquerai tout d'abord. D'autre part, cette notion ne suffit pas, ~ premiere rue, pour une @rude sur l'allure des fonctions & la fronti&re, lorsqu'on introduit les meilleures fronti~res. Aussi en th@orie classique une notion d'effilement ~ la frontiAre de Martin fur introduite et utilis@e par L. NA~M ~22~, et permit A D00B d'arriver ~ des r@sultats d6finitifs du type Fatou ; cela fut @tendu aussi A l'axiomatique des fonctions harmoniques. D'ailleurs on peut interpr@ter cette seconds notion darts le cadre le plus large de la premiere th@orie, et cette seconde th@orie peut aussi, au d@but, &tre introduite ind@pendamment darts des conditions tr~s g@n@rales.
(1) Voir un historique de la th@orie du potentiel d~n8 E2~ puis [33.
-
37
-
I. L'effilement "interne". 2. - D a n s un espace topologique semi-continues inf~rieurement sur un domaine born@ de ques
) 0
Rn
O , consid@rons un cSne convexe~de fonctions
~ 0 , contenant la fonction
+ ~ . C e l a g6n~ralise,
(ou un espace de Green), los fonctions hyperharmoni-
(c'est-A-dire surharmoniques ou
+ m ). On va indiquer des extraits de
l' @rude [ 53. La to pologie fine
~
sur
~
sera la moins fine, mais plus fine que la topologie
donn@e, rendant continues les fonctions de nage "fin" de
x0
est dit effil@ en u e ~
x0
tel que
~ . Tout compl@mentaire
•
d'un voisi-
et caract@ris@ par
U(Xo) <
lim inf
u(x)
,
xEe, x~x0 oh le second membre signifie
u(x))
sup( inf
a
(~
qui dolt @tre consid@r@ comme @gal ~ filement, oh il suffit de prendre R~duite
Re
voisinage de
Xo)
xeena
signifie
+ ~
lorsque
x0 ~
(cas partioulier d'ef-
u = 0 ).
inf u
;
ue~, u~p s u r e
R
signifie
RO .
~
Ii y a d'@videntes propri@t@s de croissance et sous-additivit@. Noter que la sous-additivit~ d@nombrable est cons@quence de la fermeture de pour l'addition d@nombrable. Un crit~re d'effilement de
e
en
x0
est
ere RI
<
I
.
o" Sile
premier membre est nul, on dit que l'effilement est fort.
Un ensemble
e ~ x0
est dit fortement ineffil~ en i
Fsup
o La topologie fine est l'est. Disons que
x0
si
.
5
r@guli~re, uniformisable, si la topologie donn@e s~pa.~e,r r~xO~ = 0 sur C{xo} • Alors , si R eI est negligeable, si R~
est d~nombrablement sous-additive, aucune suite geables ne pout converger selon ficile.
>. 1
x0
x
n
non constante de points n@gli-
~ ; ce qui rend cette topologle fine d'usage dif-
-
38
-
3. - Cependant elle a, avec la topologie donn@e, des relations importantes, qui traduisent des notions de Consid@rons,
sur un ensemble
ses valeurs darts un espace
E
non effil@ en
x 0 ~ E , une fonction
x-*x
(accumulation)
f
> ~
.
fine (fine cluster value) est valeur d'adh@-
rence sur tout voisinage fin. Les r@ciproques sont plus difficiles (B) Supposono
(b) l'effilement ~'
Alors r6elle,
(2):
:
(a) la sous-additivit@
(c)
prenant
0 :
, sur un voisinage fin, implique
Une valeur d'adh~rence
f
O'
(A) I1 est 6vident que, pour f -~
en topologie donn@e.
en
d@nombrable des r@duites,
x0
est fort,
a une base d6nombrable de voisinages de tout point. f - ~
implique
f -.~
sur un voisinage fin convemable.
lim sup fine f = lim sup
tre contenu).
sur un voisinage fin convenable
Si
f
est
(et sur tout au-
Ii y a extension A un ensemble d@nombrable de fonctions,
c'est-~-dire
avec voisinage fin commun dans les r@sultats. (C) Un r@sultat un peu paradoxal est l'interpr6tation
suivante de toute valeur
d'adh6rence fine. On suppose : (a) une base d~nombrable de voisinages de (b) l'ineffilement
est toujours fort en
x0 , x0 ,
(c) une base d@nombrable des voisinages de tout point de
Q' .
Alors route valeur d'adh~rence fine est la limite sur un ensemble non effil6 convenable.
4. - Ii y a d'autres relations importantes avec la topologie donn@e, par des th@or~mes du type Lusin. C'est une question reprise r6cemment par FUGLEDE ~173. Voici des notions tr~s g~n@rales DEFINITIONS. semble
- Un poids
p(e)
: sera une fonction r6elle
e . On dira que :
(2) Inspir@es de CARTAN et DOOB.
~ 0
croissants de l'en-
-
p(e)
est fin si
p(~) = p(e)
(~
p(e)
est d@nombrablement sous-additif si
p(e)
est continu ~ droite si
39
-
adh@rence fine de
e) ;
p(O en) ~ ~ P(en) ;
p(e) = inf p(~) ,
~
cuvert
~ e .
Quasi-partout (q. p.) signifiera "sauf s u r u n ensemble de poids nul". Un ensemble
w
est quasi-cuvert s'il existe
~'
ouvert
~ ~
avec
p(w' - ~)
arbitrairement petit. Son compl@mentaire est ~uasi-ferm.~. Une fonction que
fIC~
f
est quasi-continue s'il existe
(restriction A
~
de poids
< t
arbitraire tel
Ca ) soit continue.
Notions analogues pour la semi-continuit@ de
f
r@elle.
En th@orie classique du potentiel, la capacit@ ext@rieure est un poids fin, et on connaissait alcrs la propri@t@ suivante, que je donne darts le cas g@n@ral. THEOR~E.
- S_~i p
est fin,
f
~uasi-continue est finement continue ~. p.
La r@ciproque met en jeu une propri@t@ donn@e par CHOQUET dans le cas classique et pour certains potentiele ~ ncyaux C12~ : Pro pri~t@ de Cho~uet. - L'ensemble des points de @tre enferm@ daus un ouvert
~
tel que
Ce , ob
p(m n e) < e
e
est effil@, peut
arbitraire
> 0 , ce qui
@quivaut ~ "tout ouvert fin est quasi-cuvert", ou encore "tout ferm@ fin est quasifermi". Voici alors un th@or~me inspir@ de DOOB, et g@n@ralisant la propri@t@ de quasicontinuit@ des potentiels classiques pour le poids-capacit@ (CARTAN). TH~OREME. - supposons pour Choquet. Alors, si based@nombrable
f
p
la sgusyadditivit@ d@nombrable et la propri@t~ de
est finement continue %. p. et Rrend ses valeurs dans
d'ouverts,
f
~'
est quasi-continue.
Ce qui para~t tout ~ fait nouveau est la r@ciproque suivante : TH~OR~E.
- ~pposons
~
~ base d@nombrable,
et continue ~ droite. Sil'enveloppe tions de
~
p
d@nombrablement sous-additive
inf@rieure d'une famille quelconque de fonc-
est quasi-continue sup~rieurement, la ~r0~ri@t@ de Cho~uet est vraie.
On retiendra que, si
p
est d@nombrablement sous-additive continue ~ droite, et
& base d@nombrable, la propri@t@ de Choquet @quivant ~ la suivante : Toute fonction r@elle finement semi-continue sup@rieurement est quasi-semicontinue sup@rieurement. Sans aller plus avant (voir d'autres d@veloppements darts FUGLEDE ~16~, C17~), voici des exemples de poids :
-
~ (Xo)
oh
P
quelconque
40
-
>I 0 .
Si
F
est semi-continue inf@rieurement (s. c. i.), ce poids est fin.
Si
P
est finie-continue
> 0 , ce poids est continu A droite, et la propri~t~
de Choquet oorrespondante 6~uivaut ~ la suivante : Pour route
f >i0 , finement s. c. i. st
inf Rf_~(Xo) = 0
% F ,
(0 ..
5. - Exemples d'applioation. D'abord en th@orie classique du potentiel (espac e s de G r e e n , f o n c t i o n s On r a p p e l l e tiel
> 0
l'infini En
qu'un ensemble polaire
infini
~quivalenoe
hyperharmoniques
qu'il
de
sur lui soit
Rn , t o u t
est
(au moins).
polaire point
x E • , on dit que
~ 0 ; voir
~3~, [ 1 0 ~ )
:
caract@ris@ par l'existence
d'un poten-
P o u r un e n s e m b l e form~ dWun p o i n t ,
ou n ~ g l i g e a b l e .
Lorsqu'on n'introduit
il y a
p a s de p o i n t
&
est polaire.
e
est effil~ si
~x~
est polaire et
• - Ix]
effil~.
Un ensemble polaire est caract~ris~ par la propri~t~ que ses points sont "polaires" et finement isol@s, et l'adh@rence fine 1'ensemble (dit base) oh des points polaires de
• •
~
de
•
quelconque est la r@union de
est non effil@ et de l'ensemble (d'ailleurs polaire) oh
e
est effil@.
Un premier exemple important d'application de la topologie fine concerne l'allure d'une fonction surharmonique u
est d6finie, lorsque Cone
fine en pSle
lorsque
x0
C~
u >I 0
en un point-frontiAre
est effil@ en
x0
de l'ouvert
(ici finement isol@ sur
(log Ix _Ixol
pour
n = 2 , ou
oh
x0 . Cw ) est isol@, il y a une limite
x 0 ; et de mSme pour le quotient par la fonction fondamentale
x0
~
Ix - x0 In-21
pour
hxo
de
n > 2) .
Rappelons la caract@risation, pour le probl~me de Dirichlet, d'un point-frontiers irr~gulier par l'effilement e n c e point ou compl@mentaire, et le rSle fondamental de l a b a s e d a n s l a t h ~ o r i e DOOB [ 1 3 3 s u r l ' a l l u r e a un s o n s ) . mssure nulle, sur l'allure d'avoir
du b a l a y a g e .
du q u o t i e n t
I1 y a u n s l i m i t e
fine
quotient
une d ~ m o n s t r a t i o n
Quant a u x a p p l i c a t i o n s
s e u l e m e n t e n c o r e un r ~ s u l t a t
u/v
de deux f o n c t i o n s
finie,
sauf aux points
pour la mesure associ~e d'un tel
3ignalons
~
v . DOOB u t i l i s e
A la frontiers
de M a r t i n ,
surharmoniques
de
(1~ oh i l
d'un ensemble polaire pour cela et il
de
ses r~sultats
serait
soulmitable
plus directs. aux f o n c t i o n s
analytiques,
elles
d~rivent
des pr~c~dentes,
pas toujours de fa~on ~vidente. Signalons l'extension suivante par DOOB ~15~ du
-
41
-
thdorbmede Weierstrass : pour une fonction m4romorphe au voisina~e de l'ensemble d'accumulation fine en
z0
z0
(exclu)
est une seule valeur (limite fine) ou bien
le plan complexe entier. On se rambne au cas oh l'infini n'est pas valeur d'accumulation et au premier exemple citd plus haut. II y a d'ailleurs (selon TODA) extension ~mm4diate ~
z0
seulement finement isol4.
L'introduction des axiomatiques des fonctions harmoniques (voir [8]) qul s'appliquent aux 4quations aux ddriv4es partielles du second ordre, de type elliptique ou parabolique, conduit ~ uae extension syst4matique de ce qui pr4c~de. Tel est le cas de l'axiomatique que j'ai d4veloppde [4] avec les axiomes I , 2 , 3 , D , une base d4nombrable d'ouverts, et l'existence d'un potentiel
> 0
(cas not4 (AI)). Les ex-
tensions sont faciles, mais il faut pour quelques points, semble-@-il, beaucoup plus d'hypoth~ses (permettant l'introduction des fonctioms harmoniques adJointes de Mme HERVE [21]) (5), par exemple pour l'allure de de
u/v
U/hx0
au voisinage de
x 0 , ou
(r4sultat de DOOB). Pour d'autres, il suffit d'axiomes plus faibles (BAUER
[13) Quant au poids, dsns lee hypotheses (At) ,
R~(x0)
est, pour
~
ment borne, un poids poss4dant la propri4t4 de Choquet, et de m~me
s. c. i. locale~ R e dm , oh
m ~@
est une mesure ne ohargeant pas lee ensembles polaires. On peut remplaoer la borne locale de
~
par une majoration
~ ~ V
surharmonique, mais il faut des hypotheses
suppl4mentaires (oas (A2)) . Ajoutons encore une am41ioration d'un r4sultat, non encore publi4, annonc4 au Symposium de Berkeley
(4t4
Disons qu'un potentiel
infk RV
(w0
1965) [9].
V ~ 0
= 0
est semi-born4 si, pour l'ensemble
(4), ce qui ~quivaut &
domaine par exemple r4gulier,
w0 dPx 0
~
k dPx 0w0 ~
mesure harmonique en
e k = Ix , V > k~
0
x 0 e ~0 ) .
Dane le cab classique ( o u e n axiomatique A2) , c e l a 4quivaut d'ailleurs ~ dire que l'ensemble oh
V = + =
est de mesure nulle pour la mesure associ~e ~
Alors, dane le cab A I, si
~i ~ 0
V .
est finement s. c. s., major~e par un tel
V
(3) Tel est (A~), le cas obtenu en adjoignant la "proportionalitd (des potentiels de mSme support ponctuel), i existence d u n e base d'ouverts compl~tement d4terminants, l'identit4 de l'effilement et de l'effilement adjoint. Cela est r4alis4 pour les @quations de type elliptique dans R n ~ coefficients assez r~guliers, dont les solutions sont prises comme fonctions harmoniques. (4) On note
f(x) = i~- inf f(y) . y~x
-
42
-
fix~, et ddcrit un ordonnd filtramt d@croissant,
on a :
R
.
II. L'effilement minimal.
6. - Considdrons (5), sur un ensemble
O
d'abord sans topologie, un cSne
U
N
de fonctions finies tiels) non partout Axiomes
(a)
~ 0 ~ •
et un cSne convexe Z
P
de fonctions
sera l'ensemble des
~ 0
(dites poten-
u + p .
:
vI E Z }
-~---> inf(v I , v2) e Z ;
v 2 eZ
(b)
Ul + Pl ~< u2 + P2
==>
Ul ~< u2 ;
d'oh l'unicitd de la ddcomposition u=0
et la propri@t@ que
u <~ p
implique
.
Fonctions minimales de par
u + p
h
U .
est proportionnelle
h ~ U
est dite minimale si toute
U
Ace
que
h
Comme plus haut, on note
Effilement de
e c O
RE =
inf v
cp
v e E o u +~
(~ >~ 0)
et
R
= RO . q~
sur E
relativement A
h
minimale
~ 0 . La condition de ddfini-
# h . Elle @quivaut A l'existence d'un potentiel majorant
Noter que, pour route
h
minimale
~ 0 ,
~
est effil@,
pldmentaires des ensembles effil@s relativement TH~0R~ME. - Soi___~t h
mi-
est un point d'une gdndratrice extr@male.
v~
~
u e U , ordonnd par l'ordre na-
est convexe et est le cSne positif de l'espace, la c o n d l t i o n " h
nimale"dquivaut
tion est
major@e
fi h .
Dane l'espace vectoriel des diffdrences de deux turel, si
u e U
minimale
~ 0 , et soit
oh il a un sens, a une limite selon Ce sont les limltes selon ces
Eh
~h
A
h
O
v/h
sur
e .
non effil@. Les com-
forment un filtre
v e Z •
h
~h "
dans l'ensemble
01 ,
dgale A
inf v/h . 01 qui seront essentielles dams les applications,
zous le nora de limites fines pour la raison suivante, avec une interpr@tation topologique.
(5) Cette introduction g~ndrale abstraite est extraite de la th&se de
GOW~~
[18].
-
Supposons fonction
~
+ =
topologique, les
u
O
A1
p
s. c. i., et adjoignons la
v = u + p , le cSne
@
de la partie I. Ii y
une topologie fine.
Introduisons les classes d'dquivalence h # 0
-
continues, les
pour former, avec les
correspond sur
43
h
des fonctions proportionnelles ~ un
et pour lesquelles le filtre est le mSme et sera not@
est dit fronti~re minimale. On note O
la topologie fine et donnant A tout
restriction a
~
est
~
@
des voisinages dont la
une topologie la plus fine, et une autre la
moins fine, appel@e topologie fine sur ~
O .
sont bien des limites "fines" en
h .
~. - Applications ~ la th@orie classi~ue du potentiel (voir On reprend un espace de Green moniques
~ 0
~
et pour fonctions
et les potentiels de Green
Rappelons ce qu'est l'es~ace de Martin D'apr~s CONSTANTINESCU-CORNEA,
i
o~
s@parent
E
~
p
les fonctions har-
et la fronti~re de Martin
A = ~ - O •
E , il existe un espace compact uni-
est dense et o~ les prolongements continus des fonctions existent et
E - E .
Y0 e Q ,
G(x , y)
~0 G(x , y) dpy0(y) contenant
et
pour une famille de fonctions r@elles continues
Appliquons cela A la famille (~ parambtre o~
u
[3~, ~6~).
~ 0 .
dams un espace localement compact non compact que
®
.
Ii y a m~me parmi ces topologies
Les limites selon
. Leur ensemble
~ = ~ u A 1 : il y existe des topologies
induisant sur
~
~
est la fonction de Green et
, o~
dp
G'(x , yo )
est la mesure harmonique en
YO " L'espace correspondant
, y) K(x , y) = G'(x , y)
y ) des fonctions
YO
l'int@grale
pour un domaine
~ , inddpendant du choix de
YO ' WO
w0 ~ un
hom@omorphisme prbs, est l'espace de Martin. On note @gales a
K(X , y) 1
en
YO
ou
K~
les prolongements. Les fonctions harmoniques minimales
sont certaines
~
, et les
X
correspondants sont dits mini-
maux. On peut les identifier aux points de la frontibre ainimale toute fonction harmonique
u ~ 0 , se repr@sente de fa@on unique selon
u(y) o~
~
A 1 . On sait que
= S
est une mesure de Radon
K(X , y) ~(X) ~ 0
sur
A
(MARTIN)
ne chargeant que
AI
(du type
G 5 ).
Mais la topologie de Martin ne suffit pas pour l'@tude A la frontibre. C'est la topologie fine d@finie plus haut (6), appliqu@e ~ notre cas, qui convient. Aussi est-
(6) II n'y a ici qu'une topologie cr~te (rdsultat inddit).
@ , et elle induit sur
A1
la topologie dis-
-
44
-
il int@ressant de comparer ces topologies et de savoir que la topologie fine sur est, parmi les topologies plus fines que celle induite par dant continues les fonctions ddfinies sur
~ , par prolongement semi-continu infd-
v(x)
rieur en topologie de Martin des fonctions Green
~ , la moins fine ren-
G'(x , yo)
ou
v
est un potentiel de
>/ 0 . On a donc une interpr@tation de l'effilement minimal aux points de
A!
dans le cadre de l'axiomatique I. Cela contient, en particulier, la propridtd que la limite fine en
X e AI
d'une fonction donn~e sur
O
est la limite selon la to-
pologie de Martin mais prise hors d'un ensemble convenable efflld (au sens minimal) sn
X . L'intdr@t de l'effilement minimal et des limites fines aux points de
en dvidence dans le rdsultat fondamental de DOOB E13J : Si avec mesure associ~e monique p.p.
>/ 0 ,
Wh
v/h
h
AI
est mis
est harmonique
dans la reprdsentation int@grale, pour touts
admet une limite fine finie en tout point de
AI
v
> 0
surhar-
du moins
d~h . Ce rdsultat, dont la d ~ o n s t r a t i o n utillse beaucoup la th&se de
L. NA'~, est basd sur l'@tude prdliminaire des solutions d'un probl~me de Dirichlet dans v/h tres
O
avec des enveloppes de Perron, d@finies par des conditions-limits pour
soit darts O , soit, et c'est @quivalent, darts ~ c'est-A-dire selon les fil~Kx .
II y a diverses extensions. Au lieu d'@tre
>/ 0 ,
v
peut @tre supposde seule-
ment bornde infdrieurement sur un voisinage fin de chaque point d'un ensemble c A 1 ; alors la conclusion a lieu p. p. si, pour
h = 1 , la condition
v
d~h
sur
surharmonique et
~ . DOOB E I4J a remplac@ aus~ O , par cells que
v
est
seulement du type BLD (un peu plus pr@cis que l'existence d'une int@grale de Dirichlet finis), et l'on peut, pour
h
quelconque, traiter de m~me des fonctions
"h-BLD" . L'intdgrale de Dirichlet d'une fonction harmonique peut aussi s'exprimer au moyen des seules limites fines sur D'un autre cStd, le quotient
u/v
AI . de deux fonctions surharmoniques, ddjA consi-
d@r~ clans I, donne lieu, en le considdrant oh il a un sens, ~ une limite fine finie non seulement dams
O
la mesure associ@e A
avec l'exception signal@e, mais sur v
A 1 , p. p. au sens de
c'est-&-dire ~ la partie harmonique de
v .
8. - Cette th@orie ne sert pas seulement A donner des r@sultats g@n@raux. Elle permet aussi de retrouver une s~rie de r@sultats classiques un peu analogues, par exemple ceux relatifs aux limites angulaires. Consid@rons un demi-espace rence euclidienne
~
identifiable A
et de plus
~
O
d~n~
Rn
limit@ par un plan
P . Alors l'adh@-
est, comus dans tous les cas de frontiAre assez r@guli~re, A I = A • On montre que, pour le quotient de deux
fonctions harmoniques, la limite fine en
X ~ AI
entrains la llmite angulaire qui
-
45
-
existe donc presque partout au sens de la mesure correspondant au d@nominateur (c'est-~-dire de la mesure de Lebesgue s i l e d@nominateur est
I ). Cela a conduit
A une ~tude compar~e approfondie de ces deux types de limites [llJ. Les r~sultats classiques de limites selon la normale peuvent aussi @tre d@duits du th@or~me g@n@ral de Doob. II s'ensuit des cons@quences int@ressantes pour les fonctions m@romorphes. Ainsi, le th@or~me de Plessner dit que, pour une fonction m@romorphe darts le demi-plan, il y a presque partout sur la droite-fronti~re une limite angulaire ou bien un ensemble d'accumulation identique au plan complexe entier. On a maintenant le m@me @nonc@, qui pr@cise l'ancien, a v e c l a limite fine et l'ensemble d'accumulation fine EISJ.
9. - Ce succ~s des deux types d'effilement, interne et minimal, et une grande analogie darts les th@ories correspondantes conduit ~ los comparer. Ind@pend-mment de l'interpr@taticn d@JA donn@e de l'effilement minimal comme un effilement "interne" sur
O u AI
pourvu d'une topologie convenable (celle de
~ ) et d'une famille
convenable, il convient de faire une comparaison des notions elles-m@mes lorsque cela a un sens. Si on consid~re sont celles de re en
x0
O
x 0 ~ O , los fonctions minimales de
et en outre les fonctions
kG(x , Xo) , et l'effilement ordinal-
est identique A l'effilement minimal relatlf A
au point minimal qui correspond A
x0
enun
Q - [x0~
G(x , x0)
c'est-~-dire
sons qu'cn peut pr@ciser.
Mais les choses sont d@JA plus compliqu@es m@me d~,~ le demi-espace. II y a encore darts R n
(n ~ 3) , et pour les ensembles d'un cSne de Stolz en
X , identit~
des effilements, mais ce n'est plus vrai sans ces restrictions ; d'ailleurs un exe~ple d'effilement minimal est donn@ par le compl@mentaire d'une boule (ou disque) tangente en
X
~ la fronti~re, dans
~ . L'effilement ordinaire implique co-
pendant l'effilement minimal, mais seulement si n = 2
n ~ 3 ; il l'implique m@me pour
en un sens "statistique", et cela s'@tend A un
O
quelconque, avec une as-
sociation convenable des points de la fronti~re euclidienne et des points de
A ,
comme consequence d'~tudes plus approfondies (~6J, ~7j).
I0. - On a examin@ naturellement pour route cette th~orie (n o 7 et 9) les extensions ~ des axiomatiques avec des hypotheses telles que (A1) ou (A 2) et elles sont effectivement possibles en g~n~ral (EI8J, EX9J, E6J, [7j), ce qui contient une s@rie d'applications aux @quations de type elliptique. Ajoutons le r@sultat r@cent in@dit de GOWRISANKARAN ~20J, qui @tend le th~or~me fondamental de Doob sur les limltes fines de (D) et
Rh
u/h , en supprimant les hypotheses
ou la "proportionalit@", qu'il avait introduites darts sa th~se EISJ.
Sans pouvoir insister davantage ni entrer dans le d@tail de ce qui pr@c~de, je
-
46
-
terminerai par un r@sultat in@dit qui va prolonger en partie & ~ de
~.I
(1)
le th@or~me fi-
:
Nous supposons (A1) , la proportionalit@ (7) et les constantes harmoniques. Consid@rons une suite d@croissante de fonctions rieurement sur harmonique
W
Nous notons
Q
~ , le
lim inf
finement semi-continues sup@-
(mSme settlement quasi-partout), major@es par une fonction sur-
semi-born@e (m@me sens que pour un potentiel ~A
le prolongement de
fine (c'est-&-dire selon les sur
~n ~ 0
inf
des
u
~h ), et
~n
sur
R G'
A1
~0
y majore
inf R
~n
= R'
A1 = Q - Q
par limite inf@rieure
d~signera, pour une fonction
hyperharmoniques
fine en tout point de
U , n ° 5).
majorant
e
sur
O
e ~ o
et dont la
e • Alors
inf ~
On peut sans doute affaiblir les hypotheses, mais il n'y a pas d'extension A un ordonn@ filtrant d@croissant de
~i "
BIBLIOGRAPHIE [I]
BAUER (H.). - Harmonische Ra~ime und ihre Potentialtheorie. - Berlin, Springer-¥erlag, 1966 (Lecture Notes in Mathematics).
L23
BRELOT (Marcel). - La th@orie moderne du potentiel, Ann. Inst. Fourier, Grenoble, t. 4, 1952, p. 113-140.
[3]
BRELOT (Marcel). - El@ments de la th@orie classique du potentiel, 3e @d. Paris, Centre de Documentation universitaire, 1965 (Les Cours de Sorbonne).
[4]
BRELOT (Marcel). - Lectures on potential theory. - Bombay, Tata Institute, 1960 (Tara Institute of fundamental Research. Lectures on Mathematics, 19).
L5]
BRELOT (Marcel). - Introduction axiomatique de l'effilement, Annali di Mat. pura ed appl., 4e s~rie, t. 57, 1962, p. 77-95.
[6]
BRELOT (Marcel). - Etude comparge des deux types d'effilement, Ann. Inst. Fourier, Grenoble, t. 15, 1965, fasc. I, p. 155-168 (Colloques internationaux du Centre national de la Recherche scientifique : La th@orie du potentiel [146. 1964. Orsay], p. 155-168).
[7]
BRELOT (Marcel). - Aspect statistique et compar@ des deux types d'effilement, Anais Acad. Bras. Cienc., t. 37, 1965, n ° i.
[8]
BW~TDT (Marcel). - Axiomatique des fcnctions harmoniques. - Montreal, Presses de l'Universit@ de Montr@al, 1966 (S@minaire de Math6matiques supgrieures,
14). [9] [10]
BRELOT (Marcel). - Capacity and balayage for decreasing sets, Symposium on Statistics and Probability [1965. Berkeley] (& para~tre). BRELOT (Marcel) et CHOQUET (G.). - Espaces et lignes de Green, Ann. Inst. Fourier, Grenoble, t. 3, 1951, p. 199-262.
(7) ou settlement l'axiome
~
, pour
h = I , de GOWRISANKARAN [18].
-
47
-
[II]
BRELOT (M.) et DOOB (J. L.). - Limites angulaires et limites fines, Ann. Inst. Fourier, Grenoble, t. 13, 1963, p. 395-415.
E12]
CHOQUET (Gustave). - Sur lee points d'effilement d'un ensemble ; application A l'@tude de la capacit@, Ann. Inst. Fourier, Grenoble, t. 9, 1959, p. 91109.
E13]
DOOB (J. L.). - A non probabilistic proof of the relative Fatou theorem, Ann. Inst. Fourier, Grenoble, t. 9, 1959, p. 293-300.
~14]
DOOB (J. L.). - Boundary properties of functions with finite Dirichlet integrals, Ann. Inst. Fourier, Grenoble, t. 12, 1962, p. 573-621.
E15]
DOOB (J. L.). - Some classical function theory theorems and their modern versions, Ann. Inst. Fourier, Grenoble, t. 15, 1965, fasc. 1, p. 113-136 (Colloques internationaux du Centre national de la Recherche scientifique : La th6orie du potentiel ~146. 1964. Orsay], p. 113-136).
E16]
FUGLEDE (Bent). - Le th@or~me du maximax et la th6orie fine du potentiel, Ann. Inst. Fourier, Grenoble, t. 15, 1965, fasc. I, p. 65-88 (Colloques internationaux du Centre national de la Recherche scientifique : La th@orie du potentiel ~146. 1964. Orsay~, p. 65-88).
E17]
FUGLEDE (Bent). - Esquisse d'une th@orie axiomatique de l'effilement et de la capacit@, C. R. Acad. Sc. Paris, t. 261, 1965, p. 3272-3274.
[18]
GOWRISANKARAN (Kohur). - Extreme harmonic functions and boundary value problems, Ann. Inst. Fourier, Grenoble, t. 13, 1963, p. 307-356
[19]
GOWRISANKARAN (Kohur). - Extreme harmonic functions and boundary value problems, 2e @d. (~ para~tre).
[20]
GOWRISANKARAN (Kohur). - Fatou-Na~m-Doob limits theorems in the axiomatic system of Brelot, Ann. Inst. Fourier, Grenoble, t. 16, 1966, fasc. 2 (A para~tre).
[21]
HERVE (RQse-Marie). - Recherches axiomatiques sur la th@orie des fonctions surharmoniques et du potentiel, Anu. Inst. Fourier, Grenoble, t. 12, 1962, p. 415-571 (Th&se Sc. math. Paris, 1961).
[223
NAIM (Linda). - Sur le rSle de la frontiAre de R. 3. Martin dane la th@orie du potentiel, Ann. Inst. Fourier, Grenoble, t. 7, 1957, p. 183-281 (Th&se Sc. math. Paris, 1957).
La raise en pages et la dactylographie math~matique de ce texte ont @t@ effectu~es par lee soins du "Secretariat math~matique", II rue Pierre Curie, PARIS 5e, pour reproduction photom@canique.
-
48
-
Lois Stables et Espaces Lp. J.Bretagnolle, D.Dacnnba-Castelle, J.L.Krivine
Rappelo~ les d6finitio~ et p~pri~t@s suivantes, dues D
I=
Une application f de T x T dans R
Schoenberg ~I].
( T ensemble quelconque ) est dire de type
positif si: f (t,t') 7 = f (t',t), ~t,t' ~T ;
1° ) 2 ° )
P
?gi,J~ n 1 =
I °)
f(ti'tj) ~ i ~ j ~0'
~n,
t i~T,
~i ER.
Les conditions suivantes sont ~quivalentes. est de type n~gatif, c'est-~-dire ~ est mne application de T x T dans R +,
,
telle que :
Ct,t')=
a)
b)
}
1<. i,~<~n
2°)
e-B~
;
~(ti'tj ) 9 i
Ct,t)=o 9J ~ O, ~ n ,
est de type positif,
~
; t i e T, ~ i £ R et > l~iL.n
~> 0.
1/2 est une distance hilbertienne, i.e IT, ~1/2~
3 °)
i--o
est un espace m~trique
isom~trique a une pattie de l'espace de Hilbert. P
2 t Si ~ est de type n~gatif sur T, s i p
eat une mesure positive sur~O, oo[
telle que ( 1 ~ z -1) d H ( z ) <
~
f
, alors
(I-e -x~) x-1 dH(x)
0 est de type n~gatif sur T; en particulier,
~
est de type n~gatif, 0 ~
1
Generalisation auxEspaces L p.
A~
Caract~risation des fonctions de type positif (et n~gativ) me d@pendant que de la
norms
-
49
-
Theorem I! Soit (E, H ) un espace mesur@ tel que LP(E, p ) Pour que f(
II x
soit de dimension infinie (0 ~ p ~ co).
- y II p) soit de type positif sum DP(E, H ) ' f ~tant %me fonction con-
tinue s R +-~ R telle que f(0) = 1, il faut et il suffit que:
f-=l
sip>2
Go
-ux~ 6~(du), si p.~ 2 3 ~ S t a n t
e
f(x) = 7
une probabilit& sur R +
0 Pour que F(I} x - yllp) soit de type n~gatif sur LP(E,H ), F &rant une fonction continue :
R+-~R telle que F(0) = 0, il faut et il suffit que : F---O F(x) = 7
si
p>2
1 -u e-uxp ~ (du)
si p<2,
~tant une mesure positive sum
0 [O,Co[ telle que
~
(qg, 1)
~(du) < +~
.
D&monstration du theorem 1: a)
La condition est suffisante; soit p.<2, rig L P ( E , H ) ,
9i
1~ i ( n
~ i 6 Rt
= o, d o n c =
>
I t i ( ~ ) - ta(oo) I p g i 9a ~- o
,
d'od
1 < i,J.L n b) La condition est n6cessaire ; supposons d'abord LP(E,~ ) = i p Posons,
I
n(
/ 1''°'''
/n ) = f([l ~'11 p+,,o°* IIn I
P~ ~) (p = oo)
(p ~-~o)
-
Soit Pn la probabilit@ ~n+l
dont ~ n e s t
(~l''''_In' o) = £ n
50-
la transform@e de Fourier; comme
(~1' " ' ' f n ) ' d'apr@s le th@or@m de Kolmogorov,
il exists
une suite de variables al~atoires X 1, .... ,Xn,... ; d@finies sur un espace de probabili@ (6~, 5 ,
P) et tells que:
n( ] ~ I , . . . ,
f n) = E ( exp i ( f l Wl + . . . + f r ~ ~
La suite X i eat en dSpendance symStrique.
)) •
( Lee mesurss images sur R n sont invariantss
par permutation des indices ) I1 exists donc une ~--alge~bre ~ conditionnellement
ind~pendantes
tells que lee X n soient
et @quidistribu&es par rapport ~ ~
.
D@signons par P(co ,dx) la probabilit@ conditionnelle r@guli@re pour X 1 par rapport
.Ona,
~3
et
E
exp i (
1X1+...+
nXn ) = T ~ E q=r
exp (i
r ~ )'
soit
n
Soient ~
et
7 ( Q+" 0n a ,
v,(
+ ~ 2 (-~,-)) -- o
Doric T ~ ( f )
est ~ e n s
p.s. puisque 7,,~ est continue.
I
I
2
~( 1 --
~ ~ [~+~ 1
1
' (I * ~ ] 1
1
÷ £~ ~ ~,j~,~,~ Comme ~
1
- ~ ;~ ~ Cc ~ . 7 ~ , ~ , ~ , }
1
~
1
=
o
est continue, on en a@duit qu'on a p.s. :
1
.
-
OP~(.~) Pour p > 2 , Si p % 2 ,
e
-aI{ IP
=
exp C-aCco)I f
-
I p ).
n'est une fonction caracteristique que si a =0; d'od le r~sultat.
I-p ( ~ 0 ) ;
a(~o) = lim (I - 5o~(~))IF
de r~partition ~ ,
51
aonc a(~) est ume variable positive
d' od: Go
f(~)
= ~1(~)
=
E ~P~(~)
J
=
e - u ~ P dG(u)
.
0 Si p = oo, on a
E (S0~(;))=
E (~(~))
= ~ l ( J ), d'o~ = I p.s.
Le th~or@m est
ainsi d~montr~ pour ip ; or LP(E, H) contient, puisqu'il est de dimension infinie, um sous-espace isom~trique ~ ip, d'od la l~re pattie du th~or@m. Pour d@montrer la seconde pattie on remarque que:
I - e
B
est de type p o s i t i f ~ ~ > 0 ,
et ~--e.O
Caract~risation des sous-espace d'une espace L p, 1 W p ~ 2.
Th@or@m 2 : Soit B u n
espace de Bamach; pour qu'il existe un espace mesur~(E,p ) et une isom~trie
de B dams L P ( E , H ) , I _~p~2, il faut et il suffit que la fonction (llt - tUl) p soit de type n~gatif sur B. R@sum6 de la d~monstratin du th~or@m 2:
elle s'appuie sur la caract~risation suivante
des espace L p.
Theorem 3: Soit Z un espace de Banach sur R, r~ticul6, tel quel
a b.
llllll
=
ll II
llf + g llp >~ #If IIp
z; + fig II p
quels que soiemt les ~l~ments f, g ~ O
II fu g II P
de Z, p ~tant un nombre r~el >11.
-
52
-
Alors il existe un espace mesur~ (E, ~), tel que Z soit isomorphe, en rant qu'espace de Banach rSticulS, ~ L p (E, H ) . La dSmanstration est trop longue pour etre donn~e ici. Lemme(2): Dolt ~ une fonction r6elle ~ 0, dSfinie sur R n, telle que ~
(x - y) soit de
type nSgatif sur R n et telle que ~(Rx) = I~I p ~ (x) pour tout x e R n, t o u t ~ R ( 1 ~ p ~ 2 ) Alors il existe une m e s u r e / ~ O ,
sur la sph@re unit~ S de R n
On en d~duit le theorem 2 dams le cas o ~ B e s t ~(x) =
IIxlIp
telle que pour tout
de dimension finie n : o n prend
et l'on volt que l'application linSaire B - ~ L P ( s , ~ ) ,
fair correspondre la fonction
s-* ~x,s P
~ x~
B
sur S est une isom~trie.
Soit alors B un espace de Banach sur lequel
#Ix - yll p
soit de type n&gatif et
la f~mJlle des sous-espace de la dimension finie de B. A chaque F ~ ~ cier un espace de Banach ~
qui
, on peut asso-
= LP(SF, ~ F ) et une isom~trie iF de F dams ~
d'apr~s ce
qui pr6c~de. On pose X(F) = [G ; G £ ~ 0n en d~duit F ~ ~
, GDF)
. X(F) ~ ~
( car F E X(F)) et X ( F ) N X ( G ) = X ( F O G )
qu'il existe un ultrafiltre LL sur ~ tel que X(F) ~ TA , quel que soit
. Bolt ~/lo le quotient de ~
L F par la relation d'equivalence:
Ii est clair que A o est un espace vectoriel r~ticul~ sur R, ainsi que son sous-espace ~I
= I(fF)F~
; il exists un entier N tel que IIfFIl~ N pour tout F~
~.
Sur ~ I '
on d~finit une semi-norme en posant
Soit ~ 2
l'espace norm~ obtenu en faisamt le quotient de ~ I
~l~ments qui ~ u l e n t
par le sous-espace des
cette semi-norme. On v@rifie que le compl~t~ de A 2 est un espace
de Banach r6ticul6 qui satisfait aux conditions du theorem 3, donc est isomorphe
-
53
-
LP(E,~) pour un certain espacemesur@ (E, p). Or on peut d@finir une application i de B dans /4 I' par I fF = iF(x)
si
x ~ F
si
x ~ F
i(x) = (fF)FG ~ , avec fF = 0
En utilisant la d@finition deLL , on voit facilement que i e s t
lin@aire et isom@trique
Elle d@finit doric une isom@trie de B darts LP(E,~).
O
Le d@terminisme des fonctions ~aussiennes ~ accroissement homo~@nes et isotropes
SUr I~ • Les fonctions al@atoires gaussiennes ~ accroissements homog@nes et isotropes sont d@finies sur un espa~e de Banach B par El X(f)
-
X(g) 1 2
=
N ( l l f - gll),
o~ N est une fonction continue R +--~ R + ;f,ge B, N ( IJf - g II) @tant de type n@gatif sur
B , x(o) = o . Rappelons que P.Levy a d@sign& sous le nom de fonction brownienne la fonction X d@fimie par N(~) = t. Cette fonction est d@finie sur les scus-espace des espace LI(E, ~ ) d'apres le th@or~m 2. Rappelons aussi que d'apres le th@or@m
B = LP(E, ~ ) , N a
n6cessairement la forme: co N(t) = f
I -~e-~tP
d~(~)
,
O~p<.2
co
f
d~(~)
<
co
et N(t) = 0
si p > 2
0 D@finiticns Nous dirons que s u r u n e s p a c e
topologique T u n e
d@terministe, si l'on a : X(f) ~V(G), ~
fonction al@atoire gaussienne X est
f ~ T, ~ G
ouvert non vide de T, V(G) ~tant le
-
54-
sous-espace de Hilbert engendr@ par les variables gaussiennes X(g), Exemple!
g EG.
Un processus gaussien stationnaire analytique ( de covariance analytique )
sur R est d6terministe.
~ 6 o z ~ m J+: Soit L P ( E , ~ ) un espace vectoriel de dimension infiniet
les fonctions ~aussiennes
accroissements homog@nes et isotropes sont d@term~nlste si ~ est une mesure diffuse, et
1 < p.~ 2.
Bibliosraphie
(1)
J.J.SchSnberg
Trans.Amer.Math.Soc.,
(2)
Raceva
Soviet Tr--~lations 1961.
~u+, 1938, p.522
-
55
-
Comments on the Martingale Convergence Theorem. S.D.Chatterji
Let ( ~
, g ,P) be a probability space and let X be a Banach space. A sequence of
X-valued Bochner-integrable random variables fn on ~2 will be said to form a martingale with respect to the sub algebras
~n n=l'2'''''~n'C~n+l
(in short
fn' ( ~ n
is a martingale ) if E where E
~Ln
fn+l = fn
n >~ 1,
n is the conditional expectation operator with respect to the ~-algebra
~n"
It is known that these operators are well-defined for arbitrary Banach-space-valued integrables functions. In the following it will be assumed that the algebra (~=
~ ~n generates the E-algebra n=1 standard reduction to this case.
~.
The general case can be handled via
My main concern will be proving almost everywhere (a.e.) convergence theorems for martingales. For the sake of brevity, I shall limit myself in this talk to considering only the following statements:
($1): If
fn = E
n f
then lim fn = f a.e. (strong limit in X) n~co
($2): I f {fn' ~'n } is a martingale and the fn'S are ~ i f o r m e l y integrable ( i . e . lira
N~oe
J
I/fn j] " I{}[ fn H ' N }
lim fn = foo n~oo (S3) : If
dP = 0 uniformely in n />1 )
~hen ~fco
such that
a.e.
I fn, (~n~ is a martingale with sup n~l
E( I~fn[l) < oe then ~ foe such that
lim fn = foe a.e. n@oo ~ W o r k partially supported by Nasa Research Grant No. NsG-568 at Kent State Univers. Kent,Ohio, USA
-
56
-
In ~ 2 I shall prove that ($1) is always true. In this generality, the result is proved by other methods in Chatterji (2b) and also in A.I. and C.I. Tulcea (6). The present prove, paralleling the proof in the scalar-valued case as in Billingsley (1), is as simple
( possibly, some would wish to say trivial ) as one could wish for.
In ~ 3, I shall prove the main theorem of thi~ paper viz, that if X satisfies the following (RN) condition (RN for Radon-Nikodym) then ($3) ( and hence trivially (S2)) is valid for all martingales. The condition referred to is: (RN) z Every ~ -additive X-valued set function H on ~ o f perry that V# , the variation o f ~
bounded variation with the pro-
, is absolutely continuous with respect to P(VF~ P)
can be represented as the indefinite integral of a X-valued Bochner-integrable function. The non-negative measure V# is defined as follows: n
n
Vp(A) = s u p I ~ II~ (Ai) H ' i=I The implication (RN) ~
: AiA ~ = ~
, Ai g ~
,~ i=1
A i = A,
n~l
($3) is more general than the statements obtainable from (6).
It also follows that (S 3) is valid for reflexive X separable dual spaces X, statements explicitly made in (6). Por reflexive X, ($2) (weaker than ($3)) was proved by different methods in (2a,b) and by Scalora (5). That some condition on X is necessary for the validity of (S 2) or (S 3) ist demonstrated by the counterexample in (2a). Here a martingale fn is constructed which Sakes values in L1(0,1) and which does not converge in any sense, weak or strong, anywhere, althoug amongst other things,
IIfnll ~ 1
In ~ ~, it is shown that the (RN) condition is also necessary if ~
for all n ~I. is seperable
(generared by a denumberable class of subsets). More precisely, in this case ($2) , (S3) and (RN) are equivalent conditions.
2s Lemma
The main probabilistic tool is the following lemmas 1 z
For any martingale ~fn' ~ n
P{A;
sup k>.n o
}' if A ~ ~ n o
II fk !#> £~ " ~
sup k~n o
and ~ >
g
0 then
llfkIldP
-
57
-
The lemm~ is known and an easy consequence of the fact that
IIfnli is a submartingale.
Theorem 1. ($1) : For any space X,
E ~n f = f
lim
a.e.
(P).
n@oo
Sketch of proof: If
f is measurable
~ =
--~ (~ then (S 1) is trivial since E ~ n f = f n=l n for sufficiently large n. for a general f, ~ fg measurable ~ such that
w (II f-fll ) ~ The following obvious inequality
iIE
n f-
E
m fll ~< lie
n ~-
E
m flI+ 2sup E ~ k Ill - f II k~>l
coupled with lemma q leads us to (Sq) quite smoothly.
3:
Given a m ~ t i n g a l e I fn' O n l
Pn (A) =
, define the set-functmons ~n on C~ n as follows:
/A fn aP.
The martingale property is equivalent to the property that nP+1 is an extension °fPn oo
to
C~n+ I.
Hence for any A a S =
function H o n ~ i s
n=q~ C ~ n~ n*colim~(A) = ~ ( A ) exists. The set-
an X-valued finitely additive set-function. The set-function is of
bounded variation if f sup f l f n IIdP < oo. The main d i f f i c u l t y convergence theorems is that ~ may not b e ~
~proving
martingale
-additive. The following lemma gives a
way out. Lemma2: Let P be a probability measure on the algebra C~of subsets of a space ~ a n d finitely additive X-valued set-function of bounded variation an 5 .
where 7' ~
H
Then
are both of bounded variation and V is a finitely additive set-function
-
58
-
such that V~ (the variation of ~ ) is singular with respect to P (i.e. given ~ , ~ > O, A ~ C ~ , P(A)4g~ V~ (A') < • ) and
~ is a ~ -additive set-function such that V~ is
absolutely continuous with respect to P (i.e. given~ ~0 ~ R
~ ~ O,P(A)~@
V~(A) ~ £
The main idea behind the proof of the lemma will be sketched. One transfers P and p to the space (S,
~_1 ) where S is a totally disconnected compact Hausdorff space and
is the algebra ofclopen sets in S, ~ 1 and~
are ~ -additive on
rated by ~ 1 "
being isomorphic to ~
~-1
. It turms out that P
~-1 and hence can be extended to the ~ -algebra
~-2 gene-
(These are standard methods in this sort of work; see e.g. (3) pp.312-
13 ). On these extended measures on ~-2 apply the Lebesgue-decomposition theorem as proved by Rickart (@) and then retrace the way back through ~ 1
to ( ~ t o obtain the
decomposition indicated in the lemma. With the help of lemmn 2, I shall now prove the main theorem of this talk: Theorem 2s If X satisfies the (PAN) property with respect to P on ~ t h e n { fn' ~ n ]
with sup J n~1
!IfnIIdP < co converges i.e. ~ f ~ lim n@oo
Sketch of the proof: is an integral. ~ ,
fn = foe
Let ~
a.s.
any martingale
such that
(P)
be as before and ~ ,~ as in lemma 2.
p restricted to ~ n
being absalutely continuous with resoect to P , is also an integral
since X has the (RN) property. Let 6~(A) =~A
h dP
A ~ C~ .And
~(A) =6-n(A) =~A hn dP
& Clearly
hn
= ~
h .
Hence ~ restricted to C~ n is also an integral i.e.
~(A)
In other words,
fn
=
=
7n(A)
gn
+
=
~A gn dP
Ag~
hn
where gn,hn are also martingales with respect to
~n"
n
A e ~n
-
Moreover ~
is
~n
59
-
h and hence theorem 1 ensures the convergence of h n to a limit.
I shall now show that lim gn
=
Given 1 > E , ~ > O, find A e ~ IE P(A') + V g ( A ) < -~ .
0
a.s. (P).
( and hence A g ( ~ n o
for some n o) such that
Now P{ sup II g n l l > ~ } n,>n o
=
P~A'
,
supIlgnII>g+ P { A ; n~n o
sup II gnll > ~ } n)n o (by lemma 1)
n>.n o 1
This is clearly enough to show that lim gn = 0
$.
a.s.
(P).
In this section, the main thing is the following lemma for real-valued submar-
tingales: Lemma
~:
If {gn' G n~. is a positive submartingale with sup E (gn) n~l (A)
=
lira
gn alP, A £ (~ =
(~n
< co such that
- additive P-con-
ig a
IP>CD
tinuous set-function then gn'S Sketch of proofs
are uniformely integrable.
If gn ~ 0 is a martingale then it is easy. In g e n e r a l ~ h n a
martingale so that 0 ~ gn ~ hn and such that~hn~induces the same N . Hence the lemma.
Theorem ~: If ~
is separable then
Sketch of proof:
Let
(S2)~(RN)
. Hemce in this case
~ be generated by AI,A2,... and ~ n
($2)~
{ fn'
C~n~ induced by ~
(RN).
= the ~ - a l g e b r a
generated by A 1 ..... A n. Given a set-function ~ on ~ s a t i s f y i n g the martingale
(S3) ~
the condition in (RN),
is such that II fn]/satisfies the conditions
-
60
-
of Lemma 3. Hence (B2) implies the convergences of fn to f ~ • From hereon it is trivial t6 show t h a t ~ i s
the indefinite integral of fco"
Note: In the real-valued case the general martingale convergence theorem S 3 cam be deduced rapidly from S 1 by the following sequence of arguments:
(s I )
(I)
f
($2) --. f
because fn 1,n~formely integrable implies that ~ h k such that weakly in L 1 for some f.
Hence E ~n f E
~n f = fn etc.
~ E O ~ f weakly. But E ~ n f
nk
= fn for large n k. Hence
Next
(II) every umiformely integrable submartingale converges: this follows from (i) via the Doob-decomposition for submartingales. (III) Every positive martingale fn converges simce e -fn is a uniformely bounded semimartingale. + (IV) Am arbitrary martingale f n w i t h sup E fn ~ ® ference of two p o s i t i v e m~rtingales And
converges because it is the dif-
(III).
From here the same theorem for submartingales cam also be easily obtained.
-
61
-
References
(1)
Billimgsley; Ergodic Theory and Information, John wiley and Sons Inc. (1965)
(2)
ChatterJi, S.D. (a)
Martingales of Bamach-valued Random Variables. 66 (1960)
(b)
pp. 395 - 398
A note on the convergence of Banach-space Math. Annalen Vol. 153 (1964)
(3)
Dumford, N. and Schwartz J.T. N. ~.
(4)
Rickart,C.E.
pp. 142 - 1 4 9
Linear Operatozm.
part I, Interscience,
Decomposition of additive set functions. Duke Math. Jr. Vol 10
pp. 653 - 665
Scalora F.S. Abstract martimgale convergence theorems. Pacific Jr.Math. Vol II ( 1961 )
(6)
valued martingales.
(1958)
(1943) (5)
Bull Am. Math.Soc.Vol.
pp. 347 - 37~
Tulcea A.I. and Tulcea C.I. t Abstract ergodio theorems, Vol. 107
(1963)
pp. 1 0 7 -
124.
Trans. Am. Math. Soc.,
-
62
-
Faktorisierun~ yon Differentialoperatoren.
Herm~nn Dinges
Wenn eine positive Matrix P(i,J) kleine Elemen~e hat, i m S i n n e
~P(i,J)
~ I f~tr Jedes
i, damn kann I - P faktorisiert werden in eine untere und eine obere Dreiecksmatrix: X - P = (I - P-) (X - P+). Wenn die Indexmenge ein min~m~les Element hat, k~-n die Faktorisierung dutch iterierte elementare Matrixtransformationen bestimmt werden. Wenn in der Indexmenge eine beliebige Ordnung erkl~rt ist, wird P- und P + mlt Hilfe des dutch P best~mmten Markoffprozesses konstruiert. Die Faktorisierung ist eindeutig bestimmt. Ist A der infinltesimale Operator elmer Diffusion mit genGgend regul~ren Koeffiziemten, damn fGhrt eine ~huliche Oberlegung auf die Faktorisierung A
=
( A+
+
C ) A+
w o A+(f) • (x) nicht yon den Werten f(y) f G r y
mit erster Koordimate grSBer als erster
Koordinate von x abh~ngt. C ist ein Differentialoperator Bereiches yon A + ist noch ungekl~rt.
erster 0rduung. Die Frage des
-
63
-
Die Anzahl der Niveaudurchg~nge und der lokalen Maxi~alstellen von Gau~schen Prozessen. Werner Fieger.
In den letzten Jahren ist in verschiedenen VerSffentlichungen((1),(3),(4),(5),(6),(8)) die Frage untersucht worden, unter welchen Voraussetzungen fGr einen reellenGau~schen Prozess der Erwartungswert
der Anzahl der Nullniveaudurchg~nge und derDurchg~nge
dutch
das Niveau ~ existiert. FGr einen station~ren GauBschen Prozess x(t) gilt folgende, von Ylvisaker (8) bewiesene Aussage: Ist x(t) ein Gber [0,1] erkl~rter station~rer GauBscher Prozess mit stetigen Pfaden, ist E x(t) = 0 f~r alle t, und bezeichnet r(t) die Kovarianzf~]nktion coy(x(t),x(0)) yon x(t), so iSt der Erwartungswert
der Anzahl der Nullniveaudurchg~nge
[ 0,1] genau dAn~ endlich, wenn r''(0) existiert; wartungswert gleich
~ .
-r''(O)
von x(t) in
existiert r''(0), so ist dieser Er-
.
r Von Bulinskaya (1), Ire (3) und Ivanov (4) wurden st~rkere hinreichende Bedingungen "
fGr die Existenz des Erwartungswertes
angegeben. Da jeder GauSsche Prozess mit
var x(t)> 0 mi~ Wahrscheinlichkeit Eins keine tangentiale Nullstelle hat, gilt die obige Aussage yon Ylvisaker auch fGr die Nullstellen yon Gau2schen Prozessen. Der nichtstation~re Fall wurde yon Leadbetter und Cryer ((5),(6)) beh~ndelt. Sie fanden Bedingungen,
die hinreichend,
aber nicht notwendig fGr die Existenz dieses Er-
wartungswert es sind.
Im felgenden wird eine fGr den nichtstation~ren Fall notwendige und hinreichende Bedingung angegeben. Ferner wird dargekegt, dab alle Ergebnisse auch fGr separable unstetige Prozesse gelten, wenn man die GrSBe "Anzahl tier Nullniveaudurchg~nge"
ent-
sprechend ab~ndert. 1. Es sei f(t) eine Gber C 0,1S erkl~rte reelle ~]n~tion. Wir nennen toaEO,1S Nullniveaudurchgangsp~nkt
einen
(NND-I~n~t) yon f(t), wenn
S ( t o , ~ ) : = s u p l f ( ~ ) : ~ ( t o -£ ,t o + ~ ). [0,1]J> 0
~
J(to,~):
= inf[ f(¢): =T~(to - ~ , t o + g ). [ 0 , 1 ~ f ~ r
=
jedes
£ > 0 gilt.
-
64
-
t o ist also genau dA~n ein NND-Punkt, wenn in Jeder Umgebung yon t o ~ 1 ~ t e f(tl) f(t2) ~ 0 existieren;
tl,t 2 mit
die obige Definition ist daher nur eine andere Fassung der
Definition von Ylvisaker. Welter nennen wir t o e (a,b) einen beiderseitigen N N I > - ~ , ist und JL(to, & )
,
= in~[ f(~)
: ~ ( t
:T~
JR(to, & )
o
-£,t
(to,t o
o)
wenn entweder fCt o) >
0
•Lo,1]] < o und
+ 6 ) "~0,~3 < 0 fGr jedes £ > 0 gilt odor
f (to)~(~Ist und SL(t o, ~ ) , = sup{ f ( ~ ) SR(t o, £ )
,
=
sup
[f(~)
~ ~ (t o - ~ ,t o) .[ o,1 aj > o u~d :r ~ (to,t o + £ )
f(t o) = 0 ist und SL(to,E) JL(to,6)
0, JR(to,~)
"/0,113 > 0 fGr jedes
g ~ 0 gilt oder
0 , SR(to,~) > 0
0 fGr jedes ~>O gilt.
Es sol nun gesetzt: ~(t) =
r
1 , falls t NND-Punkt , aber kein beidseitiger NND-~,~kt yon f(t) ist,
t 0
yl
sonst , falls ~(t)
= 1 ist und entweder f(t)> 0 und ~L(t, 6) < 0
fGr jedes&> 0 oder f(t) < 0 und SL(t , 6 ) > 6[L(t)
=
t 0
g > 0
0 fGr jedes
gilt,
sonst R(t) entsprechend ~L(t) mit JR(t, g ), SR(t , ~ ) an Stelle von JL(t, 6 ),SL(t , g ) und
~(t)
s =
{I0 ' falls i S t tein 'beidseitiger s o n NND-I%n~t d t . v°n f(t)
Mit diesen GrSBen deflnieren wir die Anzahl der in (tl,t ~
enthaltenen NND-l%nkte yon
f(t) dutch m(t I ,t2; 0) := t ~ (t I ,t2)
~ ( t ) + ~R(~I) + 6tL(t2) + 2 ~
Haben wir zum Beispiel
t a (t I ,t 2)
-1
f
for
O<,t~ 1
•
~(t):=
-1
f drt
= 1
+1
fur ~ W t ~
1,
if(t) + ~(tl)
+ /~(t2).
-
so wird in m(o,1;o)
derNND-~kt
65
-
t= 1/4 einfach, der doppelte NND-Punkt t=1/2 aber
zweifach gez~hlt. Ist f(t 2) ¢ 0, so gilt m(tl,t2;O) + m(t2,t3;O) = m(tl,t3;O ). Entsprechend definieren wit die Anzahl m(tl,t2~ ~) d e r ~ k t e
aus (tl,t2S , in denen
f(t) durch das Niveau ~ geht.
2. Es sei nun x(t, ~o) # [0,1] ein Gber dem W~h~scheinlichkeitsfeld
(~
, ~ ,p) er-
kl~rter stochastischer Prozess. Man kamm zeigen, dab die Gr8Ben m( t 1,t2; 0; o~) jedes t a EO,I]
~
~(t, co), ~L(t,~o),
%R(t, oJ),
~(t, co) und
-meBbar slnd, falls x(t, ~) separabel ist. Ist p(x(t) = O) = 0 fur
, so gilt welter m(tl,t2;0; ~) ~ ~ 0,1,...,oo} m(tl,t2;0 ;o~) + m(t2,t3;O; o~) = m(tl,t3;0 ;oo)
his auf eine p-Nullmenge N~t2)E
d~ . Eine fGr o W t I ~ t2< 1 erkl~rte stochastische
Intervallsfunktion m(t 1,t2; oo) mit obigen beiden Eigenschaften kann man als Verallgemeinerumg eines Call-Prozesses
(ira Simms yon ( 2 ) )
m(tl,t2;0; o~) eine Eigenschaft,
die in etwa deSrdinarit~t
spricht
ansehen. In unserem Fall erfGllt bei Call-Prozessen
(vgl. hierzu (7)). Daher gilt als Verallgemeinerung
Em(0,1;O;~)
= supI ~
p(m(t~_1,~;
= to
ent-
des Satzes yon Korolyuk
O; o)) ~ 1), 0
~t n = I, n nat.] ;
das bedeutet also, dieser Erwartungswert
existiert genau d~nn, wenm der rechts ste-
hende Ausdruck endlich ist. Genauere Untersuchungen rechts stehendsn S~mme p(m(t~_1,t ~ ;0; ~ )
~
zeigen weiter, dab man in der
I) duroh p(m(tv_l,t~; O; ~) ungerade)
und in unserem Fall dutch p(x(t~_1; ~)x(tw; co)< 0 ersetzen kann. Wir haben damit folgenden Satz s Ist x(t, ~o) ein Gber [0,I] p(x(t,~)
= O) = 0
erkl~ter
fGr O<~t~l,
n E m(0,1;O; ~) = s u p ~ _ _
separabler stochastischer Prozess, umd gilt
so ist
p(x(t~_1,~)x(t ~ o )
= to < t 1<...
<
O) = 0 =
n nat.]
-
3.
In diesem Abschnitt sei x($, ~)
zess nit E x(t, ~)
=
0
66
-
ein Gber ~ 0,1 S erkl~Lrter separabler Gau~scher Pro-
und vat x(t, oo)
=
I
fGr o ~ t ~ 1
; r(t 1,t 2) sei die Kova-
rianz yon x(t 1, ~), x(t2, ~ ) . Die folgenden Aussagen formulieren wir mit Hilfe des Burkill-Unterteilungslntegrals: eine fthr 0 ~t' ~ t' '~ 1 integrabel mit dem
erkl~rte reelle F u n k t i o n V ( t ' , w ' ' ~
ist Burkill-unterteilungs-
(endlichen) Integralwert
~ (t',t''), wenn es zu jedem £ ~ o endlich viele l~nkte tl,...,tne(O,1) gibt,derart,da~ fGr 0 = t ov ~ t~ ~...
nit {t~,...,tm' ] D~tl,...,tnl
0
stets
Wetter man die in Tell 2 angegebene Formel fGr den vorliegenden GauBschen Prozess aus, so erh~lt man: Satz: Der Erwartungswert yon m(0,1 ; ~ ; ~ )
existiert genau damn, wenn das Burkill-
Unterteilungsintegral 1 - r(t',t'')
existiert. H~ngt das Integral
O
I - r(t',t' ')
stetig
O
von ~ ab, so gilt E m(0,1; ~ ; ~ )
=~ ~T
~2 e- ~
"~
]/__'I __ - r(t',t'')
Erkl~ren wit die Anzahl m(tl,t2; ~(t)) der Schnitt~?~-~e zweisr Kurven x(t) umd ~ (t) ale Anzahl der N N D - ~ W t e Satz: I s t ~ ( t )
von x(t) - ~ (t), so gilt:
eine Gber ~ o , 1 ]
erkl~rte reelle ~inktion m it
l~(t)J~M~ +oo, ist
x(t, ~ ) - ~(t) separabel, und existiert das Integral ~ V l
- r(t',t'') ~ , so
existiert der Erwartungswert yon m(0,1; ~(t)l~o) genau danm, wenn ~ ( t ) yon beschr~Lukter Schwankung ist. 4.
Nun untersuchen wit noch die Anzahl der lokalen Maximalstellen eines Gauaschen
Prozesses. Es sei x(t, ~ )
ein station~rer Gau~scher Prozess mit stetigen Pfaden und mit E x(t, ~o)=
= 0, var x(t,~o) = I und r(t) ~ E x(0,~)-x(t,~o)
; M(oo)
sei die Anzahl der lokalen
Maximalstellen yon x(t, ~ ) in (0,1), m(co) die der lokalen Minimalstellen. Wegen E M(co) = E m(oo) genGgt es, E (M(oo) + m(oo)) zu untersuchen. Existiert r''(O)
-
nicht,
so existiert natGrlich auch E M ( ~ )
differenzierbar,
67
-
nicht. Sind p-fast a l l e P f a d e
so ist die Anzahl der Nullniveaudurchg~nge
Anzahl der Extremstellen
differenzierbar
yon x'(t, oo) gleich der
von x(t, oo). Setzt man noch zus~tzlich voraus,
stetig ist, so erh~lt man, dab E M(co) genau d ~ n
von x(t, oo)
dab r''(t)
existiert, wenn r(t) in t=0
4-mal
ist; es ist damn
~[M(~)
+ m(~)]
I : ~"
V
/ r(4)(°) ' r"(o)
Literaturhinweise
(1)
Bulinskaya,E.V.,
On the mean number of crossings of a level by a stationary Gaussian prccess. 6(q961),
(2)
Fieger,W.,
Teor. Verojatnost.i Primenen
435 - 438
Eine fthrbeliebige
Call.Prozesse
geltende Verallgemeinerung
der Palmschen Formeln. Math. Scand. 16(1965), (3)
Ito,K.
The expected number of zeros of continuous Gaussianprocesses. 3 (1964),
(4)
Ivanov, V.A.,
121 - 147
stationary
J.Math. Kyoto Univ.
207 - 216
On the average number of crossings of a level by sample functions of a stochastic process. Teor. Verojatnost.
i Primenen 5 (1960), 319 - 323
-
(5)
Leadbetter, M.R.
68
-
On crossings of levels and curves by a wide class of stochastic processes. a~,Math. Statist. 37 (1966),
(6)
Leadbetter,M.R.
On the mean number of curve crossings by non-stationary
Cryer, J.D.
normal processes. Ann.Math. Statist.
(7)
260 - 267
Volkonakii,V.A.
36 (1965),
509 - 5 1 7
An ergodlc theorem om the distribution of the duration of fades. Teor. Verojatnost. i Primenen 9 (1960),
323
-
326 i
(8)
Ylvisaker,N.D.
The expected number of zeros of a stationary Gaussian process, Ann.Math. Statist.
36 (1965)t
1043 - 1 0 4 6
-
69
-
Vektorwerti~e Masse und Zufallsvariable auf Booleschen Algebren und der Satz yon Radon-Niko~ym.
*)
Paul Georgiou
Einleitun~t In dieser Note werden folgende Fragen studiert: I.
Es sei m eine auf einer Booleschen Algebra F erkl~rte ~ m k t i o n ,
additiv ist und deren Werte aus einem lokalkonvexen R a u m E der R a u m E
die beschr~ukt und
sind; man setzt voraus, dab
mit einer "schwachen" Topologie versehen ist und dab er vollst~ndig bezGglich
dieser Topologie ist. Mit Hilfe einer Familie yon Nikodymschen Pseudometriken und einer geeigneten FAm~lie ( F j ~
yon ~-Massalgebren
(mit strikt-positiven Massen) ist eine A
Erweiterung yon m zu einer stetigen Funktion ~ auf einer Booleschen Vollalgebra F (F g F) erreicht. Es wird dann bewiesen, u n d dab F von F erzeugt wird (§I). (reellen) F-Zufallsvariablen
dab F der projektive Limes der Familie ( 4 )
ist
Aus diesen Tatsachen ergibt sich, dab der R a u m der
der projektive Limes der F~m!lie (M ~) ist, wobei M ~ den
R a u m der (reellen) F~-Zufallsvariablen bezeichnet. Das Analogon gilt auch fGr die R~ume L I (~) umd L I ( ~ )
tier ~-und ~ -summierbaren F- und ~ - Z u f a l l s v r i a b l e n .
Dazu beweist
man, dab eine reelle F - Zufallsvariable genau dann ~-s~mmierbar ist, wenn jede "~-Projektion" 2.
yon ihr ~ -sx~mmierbar ist (§2).
Wenn (F,w) eine ~ -Wahrscheinlichkeitsalgebra
ist, dann werden die auf ihr er-
erkl~rten Zufallsvariablen mit Werten aus e i n e m u n i f o r m e n
RaumX
studiert. Es wird
auch ein Entwurf fGr eine Integration dieser Zufallsvariablen gegeben. Diese Untersuchungen werden in Zus~mmeD~qa~g m~t einer Familie von reellen F~nktionen,
die die
Struktur von X definiert, betrachtet (~3). Im Falle, dab X = E, ergibt sich m it Hilfe y o n bakannten Resultaten von Krickeberg und Pauc (vgl. (5), (6),) dab ffir jedes Mass m auf F mit Werten in E ein Radon-Nikodymscher Integrand von m bezGglich w existiert, d.h. es gibt eine W-Sl~mm~erbare Zufallsvariable f auf (F,w) mit Werten in E, so dab
~
f dw = m(a) fGr jedes a ~ F . Aus diesem Satz ergibt sich folgender
satz"
"Zerlegungs-
: jede schwache Zufallsvariable auf F mit den Werten in E, die schwach w-inte-
*) Die vorliegende Arbeit wurde durch die K~nigl. Griechische
Forschungsst.unterstGtzt
-
70
-
grierbar ist, ist die S~Imme einer w-summierbaren Zufallsvariablen und einer schwach w-integrierbaren
schwachen Zufallsvariablen,
deren unbestimmtes Integral Werte aus
dem algebraischen Komplement yon E in E' ~ hat (§$).
Bemerkun~en Gber die Begriffe und Bezeichnungen:
Die in dieser Note verwendeten Be-
griffe Gber Massalgebren bzw. proJektive Systeme bzw. topologische lineare R~ume lehnen sich eng an die Terminologie yon D.A.Kappos
(Strukturtheorie der Wahrschein-
lichkeitsfelder und -r~ume. Springer, Berlin (1960) bzw. G.KSthe (4),
N.Bourbaki
(Th~orie des ensembles, ch. 3. Hermnn~, Paris 1956 ) bzw. G.KSthe (4) an.
Im Gbrigen
verwendet man die Gbliche Symbolik.
1.
E-Masse auf Booleschen Al~ebren.
Es s e i < E ' , E ~ ein Dualsystem yon linearen R~umen Gber R, so dab die schwache Topologie s(E') auf E vollst~Ludig ist. Definition Is
Eine ~ t i o n
m yon einer Booleschen Algbra F i n E
volladditives)E-Ma~
auf F, wenn die Abbildumgenx'
(bzw. volladditive)
besch~Lukte Masse auf F sind. m heist strikt, wenn es fGr jedes
a ~ F
- ~ O~ ein a o ~ a
gibt, soda~ m(ao)
~
o m, x' a E'
heiBt additives (bzw.
O.
FGr den Rest dieses Paragraphen bezeichnet m ein additives, Man betrachtet nun auf F die uniforme Struktur ~ definiert wird, wobei ~x,(a,b) =
striktes E-Mass auf F.
m' die durch die Pseudometriken p x'
I x' o m l (a + b), x ' ~
heir yon m ergibt sich, dab die v o n ~
(reelle) additive
E', a , b ~ F .
Aus der Strikt-
m erzeugte Topologie auf F Haussdorffsc~ ist.
bezeichne die vollst~ndige HGlle yon F, die auch eine Boolesche Algebra ist. Es gilt offenbar das
Lemmn 1 s
Wenn (at) t ~T ein L~ m-Cauchysches Netz in F ist, d ~ n
ist (m(at))t~ T
Cauchysch in E. Daraus ergibt sich, da~ m sich auf F erweitern l~J~t, m bezeichne diese Erweiterumg, wie man leicht hest~tigen kann, ein E-Mass auf F ist. Es gilt nun der Satz:
die ,
-
71
-
Satz ~ : Wenn (at) t (i)
a :
ein monoton wachsendes Netz in F i s t ,
~ T
t~T
at u n d
(ii)
lim t~T
dann gibt es ein a ~ F ,
sodaB:
~(a t) = ~(a).
Beweis.
Man betrachtet auf F die beschr~nkten positiven Masse ~ ,
: Ix' o m I ,
x ' ~ E'.
Das Netz (mx,(at)) t ~ T ist dann monoton wachsend, b e s c h r ~ t
und, folglich,
Cauchysch; also ist (at) t g T ~ - C a u c h y s c h
und sei a = ~ ^ - l i m a tmt~T Daraus folgt (ii). Ausserdem ist a = ~ / a t . In der Tat: 0ffensichtlich ist a ~ k~/ at t~T t~T wenn es nun ein b gibt,sodaB a ~ b ~ ~ at, d~n~ ist fGr jedes x ' g E ' mx,(a) = t,T ~ b ) ~ mx,(a) und daraus folgt b = a.= Lim mx, (a t ) ~ mx,( t~T Corollar I :
Fist
eime Boolesche Vollalgebra und ~ ein stetiges E-Mass auf F.
(E') bezeichme die Klasse aller endlichen Teilmengen von E'. Wenn T ~ ( E ' ) , bezeichnet (F~ , ~ dem Mass ~
bezeichne
) die Massalgebra (mit strikt positivem Mass), die yon F umd von
= sup (rex, ;
kanonische Abbildung ~q
yon FV auf F~
dann
x'g
~
) erzeugt wird. Es sei ~ V
: a-~ a rood
die
(~ -Homomorphismus) von F auf F~ . FGr ~ @ a7 ( ~ , ~ g ~ (E'))
die kanonische Abbildung . Man bemerkt, dab ~
(~ -Homomorphismus) a rood ~ V --~ a mod ~
eine stetige Abbildung des metrischen Raumes
F~
auf den metrischen Raum F ~ ist. Man kann auch lei~ht best~tigen, dab (F ~ , ~ V ) c c A
ein projektives System von Booleschen Vollalgebren ist.
Es sei ~ = Lim F~ der pro-
jektive Limes dieses Systems. ~ ist offensichtlich eine Boolesche Vollalgebra (vgl. auch F.Haimo: 566 - 576
Some limits of Boolean algebras;
Proc. Amer.Math. Soc. 2,
(1951); genauer: ~ ist eine Vollunteralgebra yon
I
~ ist ( a T ) g g ~ ( E , ) £ ~ o
genau dann, wenn a~ = ~ ( a ~ )
fGr~i~t).
( ~ c ~ ) und tier Tatsache, da~ F der Kern
ist, folgt u~m~ttelbar, dab die Abbildung a-~ (a mod ~ von F auf ~
]
F~ (bekanmtlich
(E') Aus tier Relation
(im Sinne von(~; Seite 228))
)~(E')
ein Isomorphismus
ist. Es gilt nun der
Sat~, 2: A
Es sei F o eine solche Unteralgebra von ~ , da~: (i) fiir jedes ~ a ~ ( E ' ) wird F V yon der Algebra F o mod mv erzeugt (d.h. es gibt keine andere Vollunteralgebra yon F~ Gber F o mod mv ) und (ii) ffir jedes a ~ F o existiert ein ~
, sodaB a mod ~
~
0 ~ F~ .
-
72
-
Dann wird ~ von F o erzeugt. Beweis:
Man zeigt, daB: wenn F I eine Boolesche Algebra ist, sodaB F o~ F 1 g ~, dann ist
F I keine Vollalgebra. existiert, dab (x~) x
=x
sodaB
0ffensichtlich ist F 1 keine Vollalgebra, wenn
pr~ F 1
(j(E,)
g
~
~
F~ Man setzt nun voraus,dag pro F 1 = F~ f~r jedes q und
- F 1 ; damn existiert ein Netz ( ( x ~ ) ~ i n
f~r ~ g~ , das gegen (~ ~ ¢
fur jedes 7 g
~(E')
ein C g~(E')
~ (E') o-konvergiert,
F1, sodaB
da x v = o-lim ~¢~(E') x ~
(o fGr Order). Daraus ergibt sich, dab F I keine Vollalgebra ist.-
Corollar 2 : Wenn F 1 eine Vollunteralgebra yon F i s t ,
§ 2 Integration b e z ~ l i c h
sodaB F 1 D
F, dann ist F 1 = F.
einem E-Mass.
FGr diesen Paragraphen bezeichnet (F,m) eine E-Massvollalgebra, sche Vollalgebra und m ein striktes und stetiges E-Mass auf F
d.h. F i s t
eine Boole-
(E ist der topologische
lineare Raum, der im vorigen Paragraphen definiert ist.) Nun bezeichnen A bzw. A q , den Raum der reellen einfachen F-, bzw. F~ -,Zufallsvariablen und, falls ~ ~
, bzw. ~ ,
die lineare Abbildung von A bzw A ~ , auf A~ , die folgendermaBen
definiert ist:
wenn
dann ist
k
k W&(~)
(~
=
~__ i=1
X
(ai) X
bzw. i '
~
(d 9) = ~ j=l
(bj) y~ ~q
a bezeichnet den Indikator yon a ).
Auf A betrachtet man die uniforme Struktur, Definition 2 :
die aus folgender Definition erkl~rt ist:
Ein Netz ( ~ t ) t g T in A heiBt Cauchysch nach der mittleren Konver-
genz bezGglich m ( ~
~ (~,~g~(E')),
( kurz: m-Mt-Cauchysch),
wenn fur jedes 0 g J(E')
das Netz
(~t))t£ T in A T - Cauchysch nach der mittleren Konvergenz bezGglich
Die Topologie
dieses uniformen Raumes ist lokalkonvex und Hausdorffsch.
~V
ist.
Die Elemente
der vollst~ndigen HGlle Ll(m) von A bezGglich dieser Struktur heiBen m-s~imm~erbare F-Zufalssvariable.
Auf Ll(m) ist eine stetige, lineare Funktion m m it Werten in E
-
73
-
definiert, die m-Integral heiBt und yon der Funktion ~-~ wobei
k
m( ~ ) v o n A in E erzeugt ist,
k m(a i) x i
:
ai
i=1
:
@
i=1
Leicht k~n~ man best~tigen, dab die linearen Funktionen ~
, ~6 ~ , ~, 0 £ 3~(E') stetig
bez0glich der Topologien der p~- und ~J~ -Mt-Konvergenz sind. Daraus ergibt sich, dab sieh
~ bzw.
qJ~q zu den Abbildungen ~ b z w .
erweitern lassen. Das System (Ll(p~),
~
~ q yon Lq(m) bzw.
Ll(p~) auf
Ll(p~)
) ist projektiv, wie man das leicht beweisen
ksnn, und es gilt der satz
,
Ll(m)
=
Lim
Ll(po.
Man kommt nun zur folgenden Ein Netz (~t)t ~ T in A heiBt m-Cauchysch
Definition 4|
wenn fir jedes o ~ DA~
~(E')
das Netz ( W c (~t))t ~ T in A~
(Cauchysch dem Mams m nach), po -Cauchysch ist.
ist auf A eine Hausdorffsche uniforme Struktur erkliwt (Struktur de m-Konvergenz).
Die lineare Funktion
~
bzw.
~sind
stetig bezGglich der Topologien der Konver-
genz den Massen nach, wie man das leicht beweisen k~nn. Es seien nun M bzw. M~ die vollst~ndigen HGllen von A bzw. A~ bezGglich der Strukturen der m-bzw, w~-Konvergenz und~bzw.
~
die Erweiterung auf M bzw. M p yon ~
reelle m-Zufallsvariable
bzw.~
Die Elemente yon M nennt man
, Auch in diesem Fall ist das System ( M ~ , ~
) projektiv
und es gilt der Satz ~ , M
lira Ma,.
=
F o l g e n d e n S a t z kann man l e i c h t
Satz
~
beweisens
!
Die Abbildung u = lim u~ Ll(m) Injektion von L I ( ~ )
in M i s t
ein Monomorphismns, wobei u
die Gbliche
in M~bezeichnet.
Weiter gilt der Satz 6 :
Es sei f ~ M .
fau(Ll(m))
genau d ~ ,
wenn ~ x , ( f ) ~ u x , ( L l ( ~ x , ) )
fGr jedes X t ~ E I o
-
Beweis:
74
-
Die Behauptung folgt gleich aus der Tatsache, dab f ~ M genau dann, wenn
f = (f V) E ~ ( E , ) , s o d a B
f~ ~ M ~ u n d
fL = ~L~ (f~) fGr ~ @ ~ , und dem folgenden
Lemma2: f ~ ~u~
(L~(~
)) genau d~nn, wenn
z X'
.
~ ux,(L1~x,) ) fGr jedes x'g ~
des Lemm~s folgt aus bekannten S~tzen Gber reelle Zufallsvariable
Die Behauptung
auf positiven
Massalgebren (wie z.B. dem Satz yon Lebesgue der monotonen Konvergenz u.a.~ und der Tat sachen, da~ (i) die Abbildungen ~c~ ordnungstreu und stetig bezGglich der Topologien der gleic~igen
Konvergenz s i n d u n d
ist, die yon L l ~ x , )
und ~ x'~ '
(ii) die Topologie von ~ I ~ ? ) x'~ V
~quivalent mit ~ener
induziert ist.-
3 Verall~emeinerte Zufallsvariable. In diesem Paragraphen sollen einige fGr ~ ~ nStige Vorbereitungen behandelt werden. Der Allgemeinheit halber untersucht man die auf einer ~-Wabrscheinlichkeitsalgebra (F,w) definierten Zufallsvariablen mit Werten in einemuniformen R a u m X ; die uniforme Struktur
~
bezeichne
(die Gesamtheit der Nachbarscb~ften) von X. Es sei A X (bzw.M X)
der Raum der X-wertigen einfachen Zufallsvariablen (bzw. X-wertigen Zufallsvarablen); M X ist als die vollst~ndige HGlle von ~ gemz dem Masse w nach
bezGglich deruniformen S~ruktur der Konver-
(kurz: w-Konvergenz) definiert.
Es gilt der Satz 7 : Es sei u eine reelle Funktion a u f X . (i)
Dann sind folgende Aussagen ~quivalent:
u is$ gleicbm~ssig stetig;
(ii) wenn das Netz (~t)t~ T in A X w-Cauchysch ist, damn ist (u o ~ t ) t a T w-Oauchysch. Beweis: Aus (i) folgt offensichtlich (il). Man zeigt nun, dab aus (ii) die Aussage (i) folgt: Man seize voraus, dab u nicht g l e i c h m ~ i g stetig ist; d.h. es existier~m ~ ~ 0 und, fGr jedes U ~ , Netz ( ~ ( U , V ) ~(U,V) = YV
(Xu,yU) ~ U, soda~l u(x U) - U(Yu)l > ~
! (U,V)£
~
x ~j~
mit
. Man betrachtet nun das
V ~ U ), wobei ~(U,U) = xU
(U ~ V) und die Indexmenge ~(U,V)
; (U,V)~
~
x L~
und mat V ~
U~
-
75
-
folgenderma~en gerichtet ist: (U1,V1)>> (U2,V 2)
genau d~nn, w e n n V 2 R U I. Das Netz
( u o ~(U,V)) ist aber nicht w-Cauchysch, trotzdem ( ~ (U,V)) w-Cauchysch ist.Es sei v = (ui) i ~ I eine Familie von gleichmgssig ~ e t i g e n reellen ~,~Wtionen auf X. v heiBt X-adaptiert, wenn die uniforme Struktur yon X ~quivalent mit jener ist, die von ui,
igI
induziert wird.
Satz 8 : Es seien~t (ui)i 6 I eine X-adaptierte F~m~lie und ~ t ) t ~ T t = ~ ~ Xtk. Wenn die Netze (u i o g t ) t g T' k=l ark ist auch (~t)t g T w-Cauchysch° Beweis:
i g I
w-Cauchysch sind, dann
Man setzt voraus, da~ (~t)t £ T kein w-Cauchysches Netz ist, d.h.: es exis-
tieren 8 > 0, U E w ( Vk
ein Netz in AX, wobei
I%
1V { a t k ~ asl ;
und, fGr jedes t o £ T , (Xtk'Xsl) £
tund
s mit t ~
t o und s ~
to, sodaB
uCj
)> ~ " Aus der Voraussetzung Gber die n Struktur von X folgt, dab U der Form U = 9[~__I g711~ (V~@) haben darT, wobei gi = ui x u i
und Vg eine g -Nachbarschaft in R bezeichnen. Damn kann dab ein ~ o ~
{ 1,...,n~
existier~, sodab
w (Vk ~/I { atk~ as I ; (uiq ° o ~t)t ~ T
man aber leicht best~tigen,
(Xtk ' Xsl) ~
g-.l~o (VE ~)o)} ) >
~/n ; d.h., dab
kein w-Cauchysches Netz ist.-
Es sei v = (ui) i a I eine Familie von gleichm~ssig stetigen reellen Funktionen auf X. Auf A X sei folgende lineare Funktion erkl~rt w v : g%--~ (w(u i o ~ ) ) i e I ' w(u i o 6) das Integral von u i o ~ ~m~tionen
(~carts)
uniforme Struktur damn ist
~(Wv)
( 4 ,~ )-~ w( ~ u i
o ~ -u i
(Struktur derwv-mittleren
o/~ I ) ,
~l (wv) sei die von den i~ I auf Al induzierte
Konvergenz. Wenn v
X-adaptiret ist,
gr6Ber als die Struktur der w-Konvergenz. Die Elemente der ~ ( W v ) -
vollstgndigen HGlle ~ rung von w v auf ~
bezfiglich w bezeichnet.
wobei
von A X heissen w-integrierbare Zufallsvariable.
sei auch m i t w v b e z e i c h n e t .
~(Wv)
ist genau d ~ n
wenn v die Pu~kte von X "trennt". In diesem Falle kann man
Die ErweiteHausdorffsch,
X E ~ I setzen.
Dann
heiBt die w-integrierbare Zufallsvariable f w-s~mm~erbar, wenn Wv(f)~ X. 0ffensichtlich sind die Elemente yon A X genau danu w-s~,mmierbar, wenn X eine konvexe Teilmenge yon Iist.
Es gilt nun der Satz:
-
Satz 9 :
76
-
(vgl. [ 2 ~ §1, Prop. 5, S. 11 - 12~ ) • Es seien X konvexe
Teilmenge
yon R I umd v X-adaptiert. Notwendige umd hinreichende Beding~ng dafGr, dab jede wintegrierbare Zufallsvariabla w-snmm4erbar ist, ist, dab X vollst~udig ist. Beweis:
0ffensichtlich ist die Bedingung hinreichend. Die Notwendigkeit folgt aus der
Tatsache, daBI wenn (xt) t 6 T
L~ -Cauchysch in X ist, d~D~ ist (xt)tE T
(Wv)-Cauchyseh. Es sei f = b ~ ( w . ) - l i m =~
-
lim x t t eT
~
X
in A x
x t ~ damn ist aber Wv(f) = limwv(Xt) =
.-
~ Der Satz yon Radon-Niko~Tm. Es seiem (F,w) bzw. (F,m) eine strikt positive[ -Wahrscheinlichkeitsalgebra -E-Massalgebra. Auf F betrachtet man das ~ - Mass ~x' = x' o m ; x ' ~ endliche Zerlegung ~
des Einselements
reelle einfache Zufallsvariable
(~ ~')6
E'. FGr Jede
bezeichne
~,
die
k i=1
Das Netz
yon F umd jedes x' ~ E'
bzw. eine
mx'(ai) ~ a i w(ai)
, wobsi~
=
~ a i , i = 1,...,k~
, wobei die Menge der Ze~leguagem Gblicherweise gerichtet ist,
w-konvergiert gegem eine reelle Zufallsvariable ~x'
(vgl. z.B. (6; S.525)). Es gilt
mum der Satz 10 :
Es existiert eine w-m~mmierbare Zufallsvarable ~
auf (F,w) mit Werten
im E, sodab m(a) = a~ ~ dw fGr jedes a ~ F. Beweis:
Aus der Definition yon mx,
folgt gleich, dab
( ~
;x'a E°)~ , wobei E
isomorph in ~ '
dutch die Abbildung x ~ ( x ' ( x ) ; x' ~ E') eingebettet ist. Es sei hum k g die eineaohe E-wertige Zufallsvarable i~_1 ~ ai xi, wobei
Xi
=
(mx'(ai)
; x'£
-- [
EL)
i =
}.
w(ai) Aus dem Satz 8 umd den vorigen Bemerkungen folgt, dab (~ 6)~ gegen eine E-wertige Zufallsvariable ~ w-komvergiert,
die (wie aus
die relle Zufallsvariable ~x' = x' e J a
x'
o ~ dw = J a
~
(5) folgt ) w-integrierbar ist, d.h.
w-summierbar ist. Aus der Relation
~x' dw = mx,(a ) = x'(m(a)) folgt gleich, dab fitr jedes a g
F
-
m(a)
=
£ ~a ~ dw
zEs s e i f
eine schwache Zufallsvariable in E (vgl. (3),) die schwach w-integrierbar ist
wird yon F in E' ~
Lemma ~:
-
ist.-
d,h,: fGr jedes x' £ E' die
Ma,setzt
77
m
=
pr E
(reelle) Zufallsvariable x' o f
w-s[~mmierbar ist. Dann
die ~,~ktion ~ I a -~(w(x' O X a f) ; x' ~ E')
erkl~rt.
o
(vgl. (2; ~I, Prop. 1,S.9 )) ! Es ist w(x' OZaf) = ttx'(m(a)) = ~[m(a)),
wobei ttx' die zu x' biadjungierte (bitramspos~e) Abbildung bezeichnet. Beweis:
Die erste Gleichheit ist offensichtlich ( vgl.~2; §1, prop. 1, S.9]). Die
zweite ist eine unmittelbare F o l g e r u n g v o n
[ 1; Diagr~mm (27), S.71]~
da E ' @ / K e r ( t t x ')
kanonisches ( isomorphes ) Bild von E/Ker (x') ist .-
Lemma 4:
m i s t ~-additiv, d.h. ~ ist additiv umd aus arts 0 in F folgt ~ m ( a n ) = O
is E' ~ , wobei E '~ mit der yon ~ '
imduzierten Topologie versehen ist.
Beweis: Die Additivlt~t vom ~ folgt aus lim n@ ®
w (x' o ~ an
jener yon w. Wemn an4 0 , damn ist
f) = 0 fGr jedes x ' 6 E', also lim ne~
m (an) = 0 .-
Daraus folgt ~,~m~ttelbar der Satz 11: mist
ein ~-additives E-Mass .
Corollar 3~ Jede schwach w-integrierbare schwache Zufallsvariable mit Werten in E ist die Snmme einer w-sl,mm~erbazem Zufallsvariablen und eimer schwach w-integrierbaren schwachen Zufallsvariablen, deren umbestimm~es Integral Werte aus dem algebraischen Komplement vo$E
in E '~ hat.
-
78
-
BibliomraDhie .
(I)
N. B o u : ~ i
Alg~bre, ch. 2.
HermAn-, Paris 1962
(2)
N. Bourbaki
Imt6gratlon, ch. 6.Hermann, Paris 1959
(3)
P. Georgiou
Schwache Zufallsvariable auf Boolesche
6~-Algebren
(in Druck in Math. Ann. ) (4)
G. KSthe
Topologische lineare RgoAme, Springer, Berlin 1960
(5)
K. Erickebewg
Stochastische Derivierte, Math.Nachr. 18, 203 - 217
(6)
(1958)
K.Krickeberg et
Martingales st D@rivation, Bull. Soc.Math.France 91,
Chr. Pauc
455 - 5a4
(1963)
-
79
-
Toward a Theor~ of Patterns. by Ulf Grenander University
O. Summar 2.
The purpose
f o r the description
of this paper is to present
and analysis
of patterns.
serve certain simple but foundamental These are related by similarity, these concepts pure patterns description
patterns
and deformed
signs,
transformations.
For didactic
case;
The introduction
Using both
one gets a grammatical reasons the grammatical
a more general grammer of
of the mathematical
model is
examples.
What do we mean by a pattern and how do we recognize
The casual observer will immediately
and images.
can be analysed into elementary parts,
is limited to a special but typical
Introduction.
configurations,
and deformation
to the observer.
by a number of illustrative
model intended
for this model will
ones. As a result of such an analysis
will be treated elsewhere.
accompanied
I.
it is shown how patterns
a mathematical
As buildingblocks
concepts:
synonymity
of the image presented
construction
of Stockholm
one from azother?
identify as one pattern the pictures presented
in Figure I, and as another those in Figure 2.
I []
I
Figure I
A
Jq
Figure 2 What mental process mathematical properties
lies behind our recognition
description
capable
of such processes?
process;
of being carried
can it be given a precise
out on a machine?
Given a configuration
of objects
What are the
in the plane
or in the
-
space we sometimes
recognize patterns
if the physiological
80
without
and psychological
-
any apparent
mechanism behind
intellectual
our recognition
effort. Even is too formi-
dable to be within our reach for decades to come, at least it should be possible formulate
the problem in a meaningful
Naturally
we need not confine
plicated
background
timespace:
way.
ourselves
space is necessary
time point corresponds
a distribution
H o w do we recognize
to
to plane or spatial patterns. to describe
e.g. aural patterns,
of energy over frequency.
certain motions,
A much more com-
the grand-jet@
where to each
Or take patterns of the classical
in ballet
dancer? During the last 15 years The emphazis
or so, much work has been done to find recognition
has been mostly
on the algorithmic
aspect and on the statistical
problem that one meets almost all the time when attempting feature is the large amount processing
day technology. algorithm
patterns.
f o r the future. actually carried
rich experience
amount
due to a different
does not compare
in complexity
of data processing logical
in character.
certain
Starting from simple
results and seems promising
certain
with some recognition
procedures
of but could be partly
The image presented
of patterns.
to the observer
on signs and configurations
images as equivalent a pattern
of signs we
from the point of view of the as a class, perhaps very large,
Of course we can not arrive at a unique grammatical
in abstracto~
recognition
signs, which can be composed following
operations
and this will lead us to consider
of images.
it should be able to deal with fairly
better suited for the analysis
down into elementary pieces,
shall be led to consider
as time
The reason for this may be not just the
will be broken
obsezver
leading to information
to make the recognition
that the brain is capable
organization
Our approach will be grammatical
rules.
A typical
to carry through with present
This idea has led %0 some definite
out everyday by the human brain.
tremendous
recognition.
But at the same time it is clear that the sort of pattern
achieved
decision
learns from experience and becomes more amd more efficient
goes on. Exposed to a sufficiently complicated
seems impossible
To be able to do this one has attempted
adaptive:it
pattern
of data that has to be delt with,
on a scale that sometimes
algorithms.
classificatio
it has to be related to the way the observer reacts
completely
and processes
his
stimuli. The formulation logical
of such a grammar of patterns
and mathematical
set up that sometimes
intended
for general use requires
can appear rather
complicated.
a
-
To bring out clearly the ourselves
essential
concepts
in elementary
described
in another publication. the main purpose
more concrete
of the present
and synonymity
algebra.
approach
transformations
A more general
we shall limit
of a special type.
grammar of patterns
of the pr@sent paper is to introduce
the construction of the mathematical sults concerning
-
to express many notions we need in t~rms of simple and
wellknown
Although
simplicity
in this paper to similarity
In this way it will be possible
81
certain patterns.
the concepts needed for
model we shall nevertheless These results
will be
give a sample
of re-
are intended to make the discussion
and it is hoped that they will ease the reader's
effort to some extent.
2. Signs and their configurations. The buildingblock simple,
of the patterns
will be the sign. The sign can be something very
the O's and 1's in a binary sequence,
be slightly more complikated alphanumerical
symbols.
notions,
the directed
such as arcs and linesegments
the breaking
down into pieces
arrive
at elements
Denote
the set of signs by S and use s as a generic
categories
grammars
to subdivide
S (~) , where the superscript
that these different
categories
the same superscript
on the symbols
Keeping
to
symbol for any sign in S. No
of S until we come to the classification
the set S of all signs into subsets or
ranges over some domain A. To make it clear
of signs ahould be treated differently we will put denoting
specific
s fl( ~ ) 's 2 (~) 's 3 (~) and so
signs,
case of this is when the signs form a hierarc~4y
that the signs of are assumedto
lo~ger than necessary
of patterns.
it will be necessary
on. A special
making up the
that can be combined independently.
assumption will be made about the cardinality
Sometimes
Or it can
One wants to keep the number of signs as small as possible
so that one should not pursue
of concrete
edges of a graph.
S (~) are formed f r o m signs in S (~) if
be completely
this in mind,
ordered
~
~ ~
have just a single category S of signs. ~
S ~,~
so
; the categories
S (~)
oy a relation written as "< ".
let us return for the sake of simplicity
transformed b y transformation
B = (J ~A
Our approach
to the case when we
starts as follows.
from a class of transformations
~
The signs are
so that every
-
82
-
m a p s S into itself.
Definition
2.1
The t r a n s f o r m a t i o n s
2.2
A sign s I is said
in the class ~
are c a l l e d s i m i l a r i t y
transforma-
tions. Definition
transformation
~ £ ~
to be a s i m i l a r to a n o t h e r
sign s 2 if there exists a
such that s I ~ ¢ s 2. This will be w r i t t e n as s I ~ s 2 ( m o d ~ )
or s i m p l y as s I = s 2. Very
often ~
sitio
forms a 6roup if the group o p e r a t i o n
of t r a n s f o r m a t i o n s .
a) reflexive,
so that s = s for any
a n d c) t r a n s i t i v e , equivalence
relation
in m a n y ways,
2.~
Assuming
situation
into disjoint
that the s i m i l a r i t y
of b e i n g
equivalence
is t h e n an classes.
In general this can be done
it will be c o n v e n i e n t
This will be d i s c u s s e d
of signs is d i v i d e d into s i m i l a r i t y
as compo-
so that sq = s 2 if s 2 = s 1
s I = s 3. S i m i l a r i t y
class we select one r e p r e s e n t a t i v e .
and u n i f i e d method.
Definition
s ~ S, b) symmetric,
so that S will be d e c o m p o s e d
but in a concrete
is i n t e r p r e t e d
of s i m i l a r i t y than has the p r o p e r t y
so that s I = s 2 and s 2 = s 3 implies
F r o m each e q u i v a l e n c e
simple
The r e l a t i o n
as u s u a l
in a n o t h e r
to do this via some
context.
transformations
~ form
a group the set S
classes f r o m each of w h i c h we choose a r e p r e s e n t a -
t i v e b, w h i c h will be c a l l e d a template.
The s i m i l a r i t y
class c o n t a i n i n g
the t e m p l a t e
b is d e n p t e d by S b. The set of basic signs is d e n o t e d b y S B. In this w a y we 6 e n e r a t e ~-
all our signs by the o p e r a t i o n s
and b over B. Two signs
b I = b 2. The a n a l y s i s This a n a l y s i s
sI = ~ I b l
and s 2 = ~ 2 b 2
of a g i v e n sign s consits
s = ~b
where ~
ranges
over
are s i m i l a r if an o n l y if
simply in the m a p p i n g
s
~
( ~ ,b).
is unique.
By a configuration ( S l , S 2 , . . . , s n)
of signs we shall m e a n an o r d e r e d
; sic S, i = 1,2,...,n.
sequence
of the f o r m
A n y two such c o n f i g u r a t i o n s
a n d c' = (s~ ,s2,...,s ' m) are c o m b i n e d b y the o p e r a t i o n
c = (Sl,S2,..,Sn)
'~" t h r o u g h the r~le of simple
concatenation
c The
joining
c
operation
f o r the single
c' = ( s 1 ' s 2 " " S n ' S " S ~ ' 1
...,s~).
"o" is a u t o m a t i c a l l y a s s o c i a t i v e .
configuration
c = s I o s 2 o...o s n
(2.1) It is then c o n v e n i e n t (2.2)
to w r i t e
-
So far we have only free semi-group mathematical
structure
is needed
83
structure
-
for the configurations
to get something
really helpful.
To avoid complicati-
ons at this early stage we have only spoken of finite configurations. cations
it is practical
denumerably
infinite,
to introduce
to introduce
topological
rations
So far two configurations
equal,
c and c'present
the sense described.
considerations
elements tions
the observer then bases his discrimination
to given rules,
the identification
To make this definition valid, Assumption xive, b)
2.1
a)
symmetric
all images.
although they are
c, but with certain
and combined through the joining opera-
used the members
of C will oe called images.
is made so that the equality
relation
is refle-
and transitive. with c I and c'with c~ then c o c'is
In this way the joining operation
often resei~e
as identical,
we will assume t~e following.
The identification
If c is identified
To emphasize
but on more global criteria.
form. Let us express this as follow.
according
"o". To emphasize
signs
the same image to the observer although they are not equal in
By C we shall mean the set of configurations
identified
identical
It can very well happen that two configu-
The image can be crude;
in different
in the real
Now we must think of the role of the
This will lead us to regard certain configurations
2.4
but it does
signs, n = m, and these
not on the detailed form of the vector c = (CllC2,...,c n)
Definition
To deal with
c and c'have been considered
s i = s~ ; i = 1,2,...,n.
be it a human being or a machine.
presented
keeping in mind the
and keep in touch with what really happens
if and only if they have the same number of individual
observer,
or non -
has been quite easy, but we must be careful when we continue
this line of model building,
are pairwise
denumerably
to go into this now°
Up till n o w everything
world of patterns.
In many appli-
signs that make up the total configuration.
it is then necessary
not seem necessary
configurations,
and this will be done in the obvious manner,
order between the individual convergence
infinite
and a richer
is still umiquely
that the image is conceptually special notation for the image
identified with c I o c~ °
defined in C.
different I,I1,I2,
from the configuration etc.
and
~
we shall
for the set of
-
Note the important
role attributed
of the form b(.)
~
=
observers
of signs applying
may very
different
rules
are certain real valued functions
defined on some space X. The similarity
k.b(.), c
Two different
configurations
Say e.g. that the templates
b I = b l ( X ) , b 2 = b2(x),..,
-
tothe observer.
well react differently when presented o~ identification.
84
where k is any real number.
tranformations
are
Consider the configurations.
(klbl,k2b2,...)
(2.3)
o,=
Observer I will consider c and c' as identical
if
=
while observer identical
II identifies
V
more liberally by the rule that c and c' are considered
if
~ k~b~(x)
= ~ k ' ~ by(x) ,
~
xin
a subset X O d
X
(2.5)
W i t h this change of view the set of images also form a semigroup and we can use concepts from this theory. a right ( and similary semigroups
E.g.
if c
o
cI = c
left ) cancellation
o
c 2 implies
c I = c 2 we say that we have
law, and s~m~larly other concepts
may be used.
The smallest natural number n such that a given configuration form
sI
from
o
s2
o
... o
c can be written
s n is called the number of signs of the com~iguration
in the and de-
noted b y N(c). The analysis
c-A(Sl,S2,...,Sn)
is in general not unique.
in many ways such as the following Screening single and
procedure.
sign s = ~ b .
Then all that can be written,
~ 2" Among the latter some may be identical, elements.
can be attained
one, which is not very constructive
Consider first all configurations
each set of identical
Uniquenes~
as
that can be written as a
~1bl
o
T 2 b l , arbitrary
but choose one representative
~2' bl' and b2, and treat then in the same way,
images are then left on the final list and with no repetition. basis for analysis with unique
~1 from
F r o m this list strike out those that were already ob-
tained in the first step. Go ahead to those that can be written as ~Ibl arbitrary~lf
though.
results.
still too general to be of real use.
o
~2b2 ,
and so on. All finite This list can serve as
It is obvious though that our formulation
is
-
3. Formation some
)
o f patterns.
necessary
as sbove.
is the fact that two images,
lus to the observer,
-
It would be naive to take as our patterns
of our images generated
so striking
85
can be experienced
Indeed, what makes the concept of pattern although presenting
the same perceptual
as different versions
to allow for this in any theory of patterns,
is too restricted
for general use but often adequate
of the same entity.
transformations
from the group~of
T on configurations
similarity
put Tc = ( T ~ lbl , ~
~.1.
= ~s~, T s~,...,~s~) Actually
Definition
~.1.
= c'then
(~ Sl,~S2,...P~Sn)=
in the definition
the images.
These T's are called s Tmonyme
equivalence
classes
transformations.
transformation
Two images
are identified
T such that c = Tc'.
according
of our intuitive
it is quite elementary,
The
idea of a pattern.
and its somewhat
and form con-
to a given rule we get
A class of images related to each other through synonyme
matical model can only be decide after studying Anyway,
c and c'
are called s~non.Fme classes or patterns.
When these configurations
is a formalization
of simi-
transformation.
Summing up we start with signs that are generated from given templates figurations.
defined
used in 2.
to use the same set
are called s,Fnon,ymeus if there is a symonyme corresponding
T will be uniquely
condition.
= (Sl,S~,...,Sm)
w i t h the identification
it is not at all necessary
larity and synonyme
Choose a transformation
2b2,..., ~ T nbn ). The transformation
If c = (Sl,S2,...,Sn)
elsewhere.
If c = ( ~ 1 b l , ~ 2b2,..., ~ n b n ) we shall
in the set C of images if we impose ~he following
Assu~tien
It is
and which at least illustrates
c as follows.
transformations.
sti~a-
and we shall do it in a w a y that
what we have in mind. A more general approach will be described Define
simply all ( or
transformations
If this is a reasonable
mathe-
in detail a number of concrete
cases.
abstract
appearence
is easily dis-
pelled by a few simple examples. Example points,
~.1
of all straight
lines in a plane together with isolated
so that any sign s is either a straight
the similarity plane,
Let S consist
transformations
~
consist
we can choose as our templates
line or a point in the plane.
of the Euclidean
two objects,
transformations
If we let
of the
one as a given point and one as a
-
given line in the plane.
Configurations
finite number of points and lines,
86
of points will consist
polmt
signs.
In other words,
or line is presented
geometric To write
objects
with
ween images by just counting
respect
tional
images.
transformations
we could first
the image so that,say,
of the plane.
differentiate
bet-
the point of gravity in advance.
this point so that the major axis of the moments
to the points
lies in a predetermined eigenvalue
direction
is double.
to this standard form we can immediately
Then
of inertia
( a special addi-
). After the image has
distinguish
between non-synonymous
If the image contains no point we may start from the points of intersection
of the lines and go ahead in the same way with some additional the lines are all parallel. rithms,
of their con-
the number of different points and lines in the image.
rule is required if the largest
been brought
to Euclidean
points is moved to a point in the plane specified
we rotate the plane around
if they
Our pattern would then be a class of
recognition
If it has at least one point we translate
matrix corresponding
"multiplicity'
of a
the observer will not be able to decide if a given
down an algorithm for pattern
of the constituting
possible
once or several times.
congruent
of superpositions
and we shall identify two configurations
contain the same points and lines disregarding stituting
-
but the general
rule for the case when
The reader can perhaps write down more convenient
idea should be clear by now. It leads us to introduce
algoa useful
concept. Definition
~.2
From each synonyme
selected by a specified
class
(a pattern)
of images one element
rule. This image is called the prototype
should be
(for the given rule)
of the pattern. The prototype
plays the same role for the pattern that the template
Example
Let us start from the three following
~.2
templates
does for the sign.
which are all functions
on the real line, x a R~ bl:
bl(X) = 0
b2:
b2(x~". = ~ I
b3 :
b3(x)
=
for all x
for
0 ~ x~
0
otherwise
1
for
0
otherwise
x2
0 ~ x ~ x3
(3oi)
-
87
-
where xl, x2, x 3 are three given positive numbers. The similarity transformatioms shall here consist of translations x ~ x
+ h
with arbitrary real h, so that the signs
are translates of the three functions bl(X) , b2(x) , b3(x). The jointing operation shall consist in addition, but we shall only allow such configurations where the total number of b2§ and b3§ is at most five, where successive b2§ or b3§ are separated by one b 1 and where the ¢ § are such that the intervals of definition of successive b § just have one point in commom. It is natural to let the synonyme transformations be given by wram~formations of the form x - ~ ax + b, arbitrary positive a, arbitrary real b. The patterns then represent the symbols in Morse code
i b I -~ space
(3.2)
b 2 -~ dot b 3 -~ dash
where the interpretation of the image does not depend upon its location on the time axis or on the absolute time scale, but where the lengths of the different types of signs should have given and fixed ratios. The reader may wish to think about possible algorithms for pattern identification. t'- ....
-I
t- ......
'
L--
-i
L ....
_~.J
r ....
i
:
I
:
I
I
I
i
I
I
Fi~ure 3. The reader may have noticed that these two examples have one thing in common: the transformations involved operate by transforming an underlying space. This is something that happens for many types of patterns and motivates the introduction of a special nomenclature. Definition 3.3
Consider a set X with elements x upon which acts a group of transforma-
tions G with elements g. The images will consist of certain functions I = I(x) defined on X and taking values in some space; If the symonyme transformations are all of the type I(x) -~I(gx) the resulting equivalence classes of images will be called contrast
-
88
-
patterns. X is called the background (space) and C the contrast (space). Since this concept will occur frequently it may be reasonable to describe
one more
such example. Example ~.~
Let the background consist of the plane, X = R 2, the contrast be three-
dimensional space, C = R 3, and G be the group of all transformations I
x I --~ Xlo + ~ [ x 1 c o s ~ + x 2 s i n ~
g,
(3.3) x 2 -* X2o + A [ - x I s i n ~ +
x 2 coswS
where Xlo , X2o are arbitrary constants, ~ is an arbitrary angle and % an arbitrary positive number. Given a finite or infinite sequence of linearly independent functions fi(xl,x2)
in L2(X ) we shall let a sign have the form
s = (clfil,c2fi2,c3fi3)
(3.@)
with arbitrary real constants Cl, c2, c 3. Signs will be joined to one another by vector addition so that different images I(x) are different functions R 2
> R 3. If we inter-
pret the three components of I(x) as the intensities of three basic colgrs we have a model of color patterns in the plane for a three color printing process. A slight extension of the concept of a contrast pattern is sometimes natural. We may wish to identify two images I(x) and I'(x) although no g ~ G exists for wich I(x) = I'(gx), all x ~ X. Suppose that there is a g r o u p ~ o f
one-one mappings ~
: C--~C, and
we agree to say that I and I'are synonymous if there exists a pair g, ~ w i t h
gaG,
~
be e.g.
P , such that I(x) =
~I'(gx)
, all x ~ X .
In the previous e x a m p l e ~ - c o u l d
( X l , X 2 , X 3 ) - ~ ( C X l , C X 2 , C X 3 ) , c > o so that our discrimination between images would be b a s e d on relative (but not absolute) color shadings. We shall call the
~
the
contrast chan~es. Actually it is possible to view any pattern as a contrast pattern. Indeed let the background space X be the s e t , o f
images and the contrast space consist of o and 1. For
any given image I define f(x) in X = U
as 1 if x = I and as o otherwise. If T x is
introduced simply as the synonyme transformation it is clear that for some other I" and f'the two images I and I'are synonymous if and only if there exists a T such that f(Tx) = f~x), all x ~ X, so that we actually have contrast patterns. But this constraction will be artificial and of little help when ~ has high power or dimension, and we
-
89
-
would then have to look for a better construction be natural.
This question is wholly a matter of convenience.
the last one in this section, Example
~.4
Consider
the parameters otherwise;
is somewhere
and a nomdirected
xij. Here xij = 1 if the vertices
multiple
connections
by segments
of T l S
are interested
graph H described by
case we could use a single template
but no other connections neglecting
exist.
and use these as building blocks.
tify two images I and I'if there is a permutation
T
The joining
multiplicity.
only in certain graphs it may lead to more economical
The graphs below would then constitute
o
are not allowed and xii = o for all i. Let
o ~ 2 s would be just superposition
f r o m a small number of templates
example,
i and j are joined by a segment,
of E. In the simplest
s = the graph where 1 and 2 are connected
The following
in between.
the set E : {1,2,...,n}
the ~ § be the permutations
operation
if the contrast point of view should
transforming
analysis
Or if we to start
Anyway let us idenone into the other.
one pattern for n = 4.
FiSulre 4.
To bring this within the framework background
as X = E X E, the contrast
(i,j)-~(i~j')
associated
of v i e w is possible
of patterns.
To make things
where the similarity
4.
of signs,
classes,
Identification
artificial
z (...i...)T---~
of the mappings
(...i...).
This point
perhaps.
images and patterns
could be considered
as simple as possible we have restricted
and synonymity
wellknown
we would have to choose the
as C = (o,1) and let G consist
with a p e r m u t a t i o n ~
but slightly
The set up consisting
valence
of contrsat patterns
transformations
form a group.
as a grammar
ourselves
This l e d u s
to cases to equi-
and easy to handle.
of patterns.
~k-pressed in terms of the grammatical
duced earlier our next task can be formulated
as follows.
Starting
concepts
intro-
from the set ~
of
-
images
I composed
of signs from the set ~ w e
go
-
want to map ~ on the set d ~ o f patterns,
so that given any image we can decide to what pattern it belongs.
This basic decision
problem of pattern recognition
simple algorithm
capable
of handling
must be realized by a sufficiently
the large amount of data that is often present
in a pattern analy-
siso De_~nition
4.1
An invariant
function
throughout ~ and taking values parates patterns, ~(I1)
~ = ~(I),
~(gI)
in some s p a c e , i s
~ ~ ( I 2) if 11 ~ I 2 ( m o d
= ~(I),all
called a probe.
g ~ G, defined If the function
G) we speak of a recognition
se-
function
(or procedure). The examples mentioned
so far have only been used to make the reader familiar with
concepts
We n o w turn to a more 8etailed
and notation.
analysis
of two cases, both of
them simple but typical for our approach. Consider
set J of n-dimensional
real vectors I = (Xo, Xl,...,Xn_l)
images I and I'= (Xo, ~ , . . . , X n _ 1 )
are synonymous
if there exists a g such that I = gI~
where we have used the same symbol g to denote the synonyme translation
gI'=
the subscripts This means
• (Xg,
Xl+g,... , "
are understood
x~_1+g)
and say that two
transformation
and the corresponding
integer.
g, the
Additions
in
modulo n so that we need only consider g = o, I,...,
simply that we have contrast patterns
with real contrast
n-1.
and the background
space = the cyclic group G of order n. Introduce n-1
^IH
=
2~i v ~
7-
n
e
,
~
= o, 1,...,
n
(4.1)
V=o and the subsets U ~ , ~ =
1,...,
the f i r s t ~ } I such that
I2
n-l, consisting ~ o is ~
set that will be denoted by ~ o " respect
of all images I = (Xo...Xn_ 1) for which
. The Is'with ~
It is clear that images in
to G and have the form I= (xl, x , ...,Xl).
of~
~ o we seek the v a l u e ~
=~(I)
arg
e
[ o, T
~ ~
E
o forms a
are invariant
with
= o.
Given an arbitrary
. If ~
n
~o
We say then t h a t ~
Let us deal with the case when n is a prime number. find the value
= x^ 2 . . . . Xn-1 . .
image I we first
of g for which
-
when g runs through G the values
91
-
~ g run through just G as is seen using a classical
algebraic argument and the fact that n is a prime. ~ut I ~ different from zero so t h a t ~ i s
is some complex number
uniquely defined. We can then get a simple recognition
function as follows. Theorem 4.1
The real contrast patterns on the cyclic group of prime order n can be
identified through the recognition funcfion @ .
z
if I
(e
n
~W
, ~:
if I e
I~
, ~ = I ,2, ° . .i]-] )
C
~
o
and I -- (X, X,...,x)
!
I Proof.
First of all note that we have -~ (gI)F
so that
n-q 7=o
=
~,(gI) = g +
~(I)
e
2~i ~ # ~
Xv+g
= e
have the same abmolute value for e a c h ~ .
2 ~ i ~ In) W e
2~i~ IF
e
2~i-~
n
(4.3)
<
e
n
I'have the same ~ since Iv and I ~
Further
(mI-")
= e
2~iW--J£ ~ =
-2~i S F n
(mod n). Now assume that I is synonymous with I',i.e. they
belong to the same pattern, I = gI: Then I and
as
(4.2)
o,I,2,...n-I)
(gI~)p
/k
2~i~Y (I~)
(gI~)p
=
stated in the theorem, ~ is invariant.
e
n
(I'~
, all
2~ i ~ n I
.. 2~iF],(I" ) Ip = e n
., (~),
all ~ ,
(@.5)
so that ~p with oL = y(I') - ~ I ) .
= e
2~i ~ P n
A (I~)
,
(~,4)
On the other hand suppose that the recog-
nition function ~ takes the same value for two images I and I: T h e n ~ =
e
p
(4.6)
~'
and
-
Then I = ~ I ,
92
so that I and I'aresynonymous.
--
We have several times implicitely used the
fact that the vectors v~ =o, 1 ,..., n-1
=
f[ e 2~i ~n~
, ~ = o, 1,..., n-1
I
(4.7)
_
are linearly independent and hence span the whole R n.
The case when p is not prime will be left to the reader. The reader may have noted that we have used the characters of the cyclic group in our analysis. Fourier analysis may be useful for other conmrast patterns where the background space forms a group and the synonymic transformations can be identical with the seperate group elements; it all depends upon whether the resulting algorithm is simple enough to be of use. If d ~ consists of only some of the possible patterns,
identification may be possible
through simpler recognition functions. We can then look for other probes,
linear
n-q
,.#(I) =
~ "9--
x.,,
,
(4.8)
0
or quadraic n-1
?(T)
=
>-7
x
v,p= o % - #
x
v
(@.9)
/,.,
and so on. A useful probe function could be the form
~(I)
= (
n-1 ~ ~2 :
x~, 0
n-1 2 ~ xw, ~ :0
n-1 ~ "~ :
x xv+ 1 ''''' 0
n-1 ~-~:
x.~x +p )
,
~r~ p~,.,.,n
0
(4.1o) The analogous is true for contrast patterns with other backgrounds. Let us now turn to the second case mentioned in Example 3.4, where non-directed graphs over n vertices form the images. An image
I
can be described by the ~
n~mbers
Iij; i = 1, 2,..., n , j = i+I, i+2,..., n, where Iij = I or o depending upon whether there is a segment or not joining the two vertices i, j. The recognition procedure can be put in algorithmlc form as follows. Theorem 4.2
Introduce the function a(I) defined for every image
I = ~fIi~;~
i = 1, 2,..., n;
j = i+1, i+2, ..., n~j
(4,,11)
-
93
-
by
(n-1)(n-2) 2
...+2
- I I23 + 2
(n-1)(n-2) 2
_ 2 12~- + "'" + In-1 n
(4.12)
n(n--1 ]
~hen a(1) t ~ e s to each value
the values o, I, 2,..., 2--'t-~ - I corresponds ~(1)
exactly one image.
:
max
Further,
on the 2 different images so that define for each image I the number (4.13)
a(I ° )
all I'~ I (mod G) Then d~(1) is a recognition
Proof.
Obviously
d~(1) is invariant.
f o r which a(1) = a(l').
~
function.
-I
two images I
= {Iij }
and
I'=
c,~i~j)
Then
~ dll + 2
Consider
-
2
-
(n-1)(n-2) d12 +...+ 2
-I
2
d23
+""
+ ¢n-1
n = o (4.14)
where dij = Iij - I~j is equal to 1,o or -1. Since the coefficients powers
of 2, it follows that d11 = o since the sum of the remaining terms
2
-I in absolute value.
Ii= I~ The mapping between
~
Inthe same way the other differences
and the numbers
F o r any pattern m let I m be the (unique) max a(I) = I ~ m
which ~ ( m ) =
g,
=
~6(m')
I s and I m
the patterns
=
o, 1, 2,...,
2= ~ - ~
dij = o so that
-I is hence one-one.
(4.15)
of the pattern m. Two patterns
g-1 g'l~,
so that the two prototypes The quantity ~ ( I )
analysis
the image into a special form. will be called standardization
I--~(m,g)
of the image.
that
I m and I~ are synonymous:
is a recognition
function.
In these two theorems we seek, when the image I is given, the synonyme
be attempted,
m and m'for
then have a(gI m) = a(g°I~) which we have seen implies
m and m'coincide.
g which brings
is at most
image realizing
max a(g!m). g e G
Here we can consider I m as the prototype
gl m
are decreasing
This sort of procedure, of the image.
transformation which will often
It leads to a grammatical
-
94
-
The following example with n = @ will be instructive. Let us separate J Z i n t o Mr, r = 0,1,2,3,@,5,6 where r = r(I) is the invariamt r(I) =
~--
Ii~
(4.16)
i< j F o r each Mr we divide it into its patterns,
choose from each pattern the image I = I m
w i t h largest a(I) and use it as a prototype• We get the following table.
r
number of
number of
patterns
images
patterns
value of or
0
~=0 6
•
i
or= 32 15
d.= 33 2O
V< ~=
52
o6= 56
~=
51
or= 60
%=
62
i=
63
15
6~
We can easily calculate the values of a(1) for the different images.
=50
-
95
-
Prototype of values e f a
values of~6
pattern
/ 7
~
0
0
1,2,4,8,16,32,
32
12,18,33,
33
3,5,6,9,10,17,20,24,34,36,~0,@8,
~8
7,25,42,52,
52
11,21,38,56,
56
13,14,19,22,26,28,35,37,41,~,49,50,
50
30,45,51,
51
15,23,27,29,39,43,46,53,54,57,58,60
60
31,47,55,59,61,62,
62
63,
63
The calculation of the recognition fumction
~(I)
could be implemented as follows.
If r ~ (~) go to the dual graph: Iij --~ 1 - Iij. Take one after another of the r segments as
(1, 2). Consider the set of segments startimg from mode 1 •
not empty choose one after another of theme segments
If this set is
as (1, 2). and continue in this
way. If the set is empty take one segment as (2, 3) and continue with all possibilities always preferring combinations for which the corresponding power of 2 is as high as possible. While this algorithm is general one way note that if the set I of admissible images is not too large simpler probes may be used to build up a recoo~nition function. Such probes are the simple invariants:
-
rCl)
= number of segments
96
-
of I
"
"
isolated nodes
=
"
"
nodes f r o m which exactly one segment
N2(1) =
"
"
nodes from which exactly two segments
~(1)
That they do not suffice generally
starts. start.
is clear f r o m figure 5 which shows two non-synony-
mous images I and I' w i t h r(I)
=
r(I')
=
6
No(l)
= No(I')
=
0
~I(i)
= ~I(I,)
=
s
~2(i)
= ~2(i,)
=
2
/
/
\ I
I ~
Figure 5
~.Proximit~
of patterns.
Unfortunately
most pattern recognition
images presented
in practice must cope w i t h the case w h e n the
to the observer are not identical with the original
a pure image I we are given a deformed image I~= dl, of deformation meet a space ~ formation, ones
, ~
transformations. consisting
as will be assumed throughout,
images.
points. Am before we have the s e t .
shall sometimes use the notation
J%P
all pure images I we now
If~includes
the identity trans-
the pure images from a subset of the deformed
c ~ ~ • It will be natural to introduce
G-equivalent
where d is an element f r o m a set
Instead of the s e t , o f
of all deformed
ones. Instead of
the subsets J ~ o f
~consisting
of pure patterns;
for clarity we
instead of J %
•
of
-
97
-
Figure6. The left picture of Figure 6 is a fingerprint,
the middle one synonymous with the first
under translation rotation and uniform stretching. These are pure images. In the right picture we have lost a part of the fingerprint, A recognition function should map ~ i n t o J ~
it is a deformed image.
p, but what recognition function should be
considered best? In section 6 we shall see that this problem can be posed in a meaningful way, and sometimes even solved, when the deformations are statistical in character. But at present we do not assume that we have access to amy systematic way of describing one deformation transformation as more likely, better, or preferable in any precise sense to another. I n certain p a t t e r n a n a l y s i s an image I ~ ,
situations we have nevertheless an idea of proximity:
is closer to one image I ~ ,
than to another I ~ .
Behind this feeling may
hide the fact that different d's require different effort to carry out; effort could even be as simple as physical energy.
The author believes that this idea can prove
helpful in descriptive pattern analysis: when we have to construct a mathematical model intended to describe the working of a recognition mechanism,
say a psychological-physio-
logical one. This is in contrast to normative pattern recognition when we are asked to design a recognition function that is best in some given sense. An example may be helpful, it could be extended considerably. Let us consider contrast patterns in a Euclidean space X with values in some arbitrary contrast space. Two patterns are said to be synonymous if they are congruent with respect to translations.. Let the deformations d be given b y d:
x --~ d(x),
where d(x) =
= f(xl,x2,...,x n) is (strictly) increasing in each component xV, v = I ,2,... ,n, and maps X onto X. Assume that for any d the difference x - f(x) is bounded,
II x - d(x)ll%cf
-
98
-
Let us define the norm of a deformation txansformation as
Ildll
=
inf
sup
g~G
x~X
lid(x)
- g ( x ) ll
,
where g is a translation g = (g1' g2' "''' gn ) :
(5.1)
x? ~ x w + g? .This norm (or pseudo-
norm rather) is defined throughout~T . If d is just a synonymity transformation,
d e G,
the norm vanishes, lld #~ = 0. The converse is essentially true but only under certain topological d2 ~
restrictions which shall not be discussed here, Let us note that if dl,
we have
II~ d2 ]I : i ~
sup
g&G
x~X
.~ inf ~g~G
sup x6X
.~ inf y~G
sup yaX
IIdI [d2(~)J - g(~) II {lJdlfd2(x)] II ~ ( y )
- ~[%(x)]l I
-~'(y) I~ + inf heG
: lld~il + lid211 •
+
sup x£X
JldR(X) - h(x)II
=
(5.2)
We could consider a recognition rule, minimum deformation recognition , like this: given an image I ~ a n d ting ~
= d~m~
a set~
, m~ ~ P
p of pure patterns we consider all possibilities of writ, choose the pattern ~
possible. Assuming that the minimum norm
for which
II d~l~
is as small as
l~d~ II is attained for a unique m~
we get a
recognition rule of a sort that may be of descriptive use. This may remind the reader of the many extremum principles in physics,
say the principle of least action. Only
empirical studies can decide if this approach is worthwhile. An alternative starts from some G-invariant norm contrasts on a background.
""]/I°~'/I defined
on the set of admissible
Two important cases are the uniform norm
IIz~ll
= sup/Jf~ (~)I/ u
(5.3)
X
and the Lp -norm
lJz~ll p =
(~ llz~(x)llP ~ ( ~ ) ) Y P
,
p
>~ 1
,
(5.4)
where the integral is taken with respect to a G-invariant measure. In both cases o b viously
Jlgz~l/
=
I/I~ll.
- 99 -
Theorem 5.1
Let
IlI~l]
be a G-invariant norm defined on the set ~ r o f
formed contrast images. Define for any I ~
admissible de-
, I2~the quantity
gfl'g2 e G Then d = dist(Iq ~r , I2xr ) is a distance depending only on what patterns ml~ that the deformed images belong to,d = a)
~ (ml~ , m ~
b)
~ ~
)~
~ (ml~, m2~),
0 , = 0 if m
~ , a~ ~" ) ~ A ( m l ~ ,
and
= m2~
m~'~ ) +
and m ~
(5.6)
~ ( m 2 '~ , m 3'~ )
(5.7)
Proofz For arbitrary synonyme transformations ~1 ' ~ 2 ~ G we have
gfl'g2
= dist ( , f
hfl,h2 ~ G
,,~.
But this implies that the distance is unchanged when I1~r varies over m~~ and I2~
~.
so that
~(~ ,~
) is well defi~ed o~
~
~
Prop,~y
over
a) is obvious
and b) follows from
dist (ff,zT>_- i~
II
gq, g3 eG
l'
.
Tf
_
dist (I~ ,~2 )
+ dist (I~ ,I3
, f11)
+ II
-
).
(5.8)
gq 'g2'g3 ,EG
=
This would give us a possible recognition rule, minimumdistance recognition: when the observer looks at I ~ h e will classify it as the deformed pattern m ~ a n d
search for that
pure pattern m making zh (m~T,m) as small as possible.
6. Deformed ~mages.
We have already usedcthe terminology of the following Definition 6.q.
Denote by~[ a set of transformations d defined i n ~
and having images
as values. The transformations d are called the deformations and the resulting values I~
= dI the deformed images.
-
The range ~
100
-
of the d's need not be identical with ~
; usually~
~ contains
~
and is
and especially
in the
m u c h larger. This concept plays a fundamental
role in the analysis
case when the d's are given statistically.We lity measure Example
of subsets o f ~
.
Consider the contrast pattern w i t h the plane as background X = R 2, w i t h
real contrast C = R 1 and with the synonyme plane.
must then assume that there is a probabi-
defined on a suitable ~ -algebra
6.1
of patterns
transformations
A given image I(x) will be deformed according
G = all rigid motions
of the
to the rule
I(x)--~ I(dx) where,=
all non-singular
linear transformations.
should then be defined in the 6-dimensional In this example ~
happened
Instead a deformation
6.2
plmg
d may very well destroy a part of the semantic
x --~ d(x)
and with independence happen to be different
=
of a typical
content of the
example. let the contrast be real valued and G =
of X with the transformation,
d
P
space~.
Let X consist of all integers,
all translations
distribution
to form a group but this should not oe thought
pure image as in the following Example
The probability
written as gx = x + g. Let d be a map-
f
x - I
~
x
"
"
I - 2p
x + 1
"
"
p
for different
with probability p (6.1)
values of x. If d(x - I), d(x) and d(x + 1) all
from x we will loose knowledge
about the particular value I(x)
that we would have if we knew the pur image I. Another way in which information
is lost ~ c u r s
so often that it deserves
a name of its
o~wno Definition changing
6.2
Consider
contrast patterns
and let
consist
of transformations
a given image I(x) into a deformed image I(x), I,(x)
x e Ed,
=
undefined,
~ x E Ed
(6.2)
-
101
-
where E d is a subset of X depending upon d. Such a deformation will be called a mask. Although
o~
= a group is f a r f r o m general
case of interest group of ~
is w h e n ~ i s
some group and the synonyme
group G is a (normal)
sub-
.
The following will be formulated image I = I(x) and applying formation
it occurs now and then and then a particular
in terms of contrast patterns.
the deformation
d we get a new image I ~ x ) .
g is applied to its argument we get g I ~ =
first applied g, I(x) --~I' that we get a different
= I(gx)
result,
Starting from a pure If the trans-
I(8~) = g~I. If instead we had
and then d, I'--~dI'
= dgI,
it can very well happen
gd~ ~ dgI° If this does not occur the discussion will
be simplified. Definition mute,
6.~
If, for every choice of g ~ G, d a ~ ,
gd = dg, we shall say that the deformation
This condition
is quite restrictive
the two transformations
is covariant
com-
(with respect to G).
and we will need a weaker one that holds more
often. Definition
6.4
If, for any I ~ ,
is equal to the probability
distribution
measure P of the deformation are covariant Choosing
is covariant
between competing
(with respect to G) or that the deformations
decision procedures
nable to require for consistency,
that can be used to i~entify the under-
deformed image, which one should we prefer? process
to classify a deformed image I ~ as originating that gI ~ ,
same way as coming from the pattern
idea. Assume,
of dgI ever
of gdl we shall say that the probability
should we ask that a good recognition
first glance
distribution
in probability7 (with respect to G).
lying pattern behind a presented perties
g g G, the probability
should have? If we have decided
from a pattern m it may appear reaso-
for any g a G ,
should be classified
m so that the procedure
this may seem quite convincing,
What pro-
should be G-invariant.At
but so does the following
for some pure image I, that we have recognized
in the
alternative
the deformed
image dI
as coming from a pure pattern m. ~Then we may require that any dgI, with the same deformation
d, but w i t h arbitrary g ~ G, should be classified
same pure pattern m.
as originating
from the
-
Definition 6.5
A recognition function
invariant (G) if
~(gI ~)
(u~) if
~(dI) for all d e 2
~(dgI) =
=
~(I ~)
102
-
~(I ~ ) mapping $ ~ into ~
for all g a G, I ~ 6 U
will be called
&~. It will be called invariant
, g a G, I @ ~.
A straightforward modification may be called for sometimes. Say that there is a subset D O of the deformations of probability zero, P(Do) = o, such that in its complement D ~ 0
it is true that (~)
~(dgI) =
~(dI) for all g and I. We could then speak of invariance
almost certainly.
Theorem 6.q. If the deformations are covariant with respect to the synonyme transformations
it
implies that invariance (G) is equivalent to invariance (#Z). Proofs
If the deformations are covariant (G) we can write ~ (dgI) = ~(gdI)
is invariant (G) it follows that ~ (dgI) = ~ (dI); it is invariant ( ~ ) . hand ~ (gdI) = ~(dgI)
and invariance (u~) implies
~(gdl) = ~(d~);
so that if
On the other
is invariant (G).
Theorem 6.2. If the deformations form a group ~
with the set of synonyme transformations as a normal
subgroup, invariance (u~) implies invariance (G). Proofx find~
If ~ is invariamt L/Z) it follows that, for any g ~ G, d E ' a n d
Ie ~
we can
G so that
~(~az) = ~ ( d z Z ) = since for amy
d ~
~(az)
(6.3)
and g in the normal subgroup G the element
~ = d -I gd
G.
It is easy to find examples where the two concepts of invariance do not coincide. Example 6.~
Let the background X consist of all integers, the images be sequences
~I(x),x . . . . -1,0,1,... 3 of real numbers with only a finite number of values different from zero. Let the synonyme transformations
consist of translations I ( x ) - ~
I(x + g)
and let the deformations be defined as dI(x) = d(x)I(x) where d(x) is a real sequence. If we base our classification on the probe ~ (I) = ~
I(x) the procedure is clearly
X
invariant (G) but not necessarily ( ~ )
since ~ ( d g I )
= ~ X
ferent from
~ (dI) = ~ X
d(x) I(x).
d(x) I(x + g) can be dif-
-
When do (non-trivial)
103
-
invariant recognition procedures exists and what form do they
have? If ~ s h o u l d be invariant (G) it should be constant on the subsets of ~ c o n sisting of mutually synonymous (G) deformed images. Denoting these synonyme sets by ~ where cg is a suitable chosen subscript, we will have to find out if the partition ~ =
~-) ~ i s
sufficiently fine to be able to support a sufficiently good recogni-
tion function. If there are only a few (perhaps just a single one.) syncnyme sets we may have to sacrifice the property of (g) invariance. Similarly, (J() it should be constant on s e t s ~
if ~ (I ~) is invariant
; two deformed images belong to the same set if
they can be written as dll,dl 2 with the pure images 11 and 12 belonging to the same = ~U ~
pattern. We get a partition ~ to a single set ~ ( ~ ) . non-pathological Example 6.4
(~) ~ note
that several patterns m can give rise
That degenarate cases can
occur in situations which appear
is easiest shown by an example.
Let X consist
of the integers and let C = R1; g should mean a transla-
tion by an amount g of X, G = X. Given ~ images II(X),12(x),...,Ig(x),
none of which is
synonymous to another, we get ~ patterns ml,m2,...,m ~. Assume that each sequence I ~ x ) has only a finite number of values different from zero. Deformations are defined by
l(x) - *
z(x) + d(x),
(6.~)
where d(x) is realvalued, and the probability measure P on ~ is introduced by the assumption that d(x) forms a certain stationary and ergodic process. To fix ideas let the d(x) be independently normal mean zero and standard deviation one. To discriminate between the possible ~ patterns we divide ~
into ~ disjoint subsets ~ ,
and decide to classify the deformed image I ~ a s
~,...,
coming from the pattern m ~ if I J ~
= 1,2,..., ~.To make the recognition procedure invariant (G) we ask that g ~ f - ~ for any g e G and ~ = 1,2,...,~,
but it is
~ J~ ; ~
well known that an ergodic process can
only assign the probabilities 0 or 1 to an invariant subset of the sample space. That means that the stationary ergodic stochastic process d(x) with probability measure P assigns the probability 0 or 1 t o ~ j .
But the process I(x) + d(x) can easily be shown
to have a probability distribution Pz absolutely continuous with respect to the earlier measure P. Hence P 1 ( ~ I I
from pattern m#) is either equal to 0 for all q or equal to 1
for all ~ . Hence we must go beyond the class of (G) invariant recognition procedures in our search for a good decision rule. - The following observation has been made many times but should be of interest in this context. Divide the sample space into classes
-
... K_I,K0,KI,...
104
-
in such a way that from each set
~gI,g ....
-1,0,1,... J
,for any
given I, we assign one element to each KV. We need not bother about the possibility that some I's are periodic,
gI = I for some g ~ 0, since all these I's have total pro-
bability zero. Then the K~'s are translated of each other and should have the same Pmeasure (if the are measurable).
But this measure c a ~ o t
be either
zero or positive
and we arrive at a contradiction. Hence the classes are not measurable. Let us separate three cases depending upon how much information we have about the frequency of occurrence of pure images and patterns. Case I. A probability distribution Q is given over the set ~
of pure images.
Case 2.
A probability distribution Q is given over the s e t ~
of pure pattermm.
Case ~.
No probability distribution over images or patterns is known.
Case I is the simplest to handle but is seldom a realistic assumption, handled in principle under fairly general conditions,
case 2 can be
case 3 needs additional criteria
for leading to an optimal recognition procedure. Let us at first
take a look at case 2
To avoid ~1=~ecessary trouble let us for the moment assume that we have a finite number of patterns ml,m2,...,m ~ . We will look for a p a r t i t i o n ~ = _m s e t s 6 7 ~ and use the recognition rule: If I ~ a ~
~
~
into disjoint sub-
=I
we shall assign I ~ to pattern m ~ . To measure the goodness of this decision
rule let us consider the probability of a correct decision. P = P(I ~ is recognized as belonging to m if the pure image belomgs to m) =
~ ~=I
Q (m~) P ( # ~
~is
generated by pattern m ~ ).
But this way of expressing the probability is not legitimate without further justification. Indeed the conditional probability P ( U ~ I I ~ is generated from the pure image I ) is not in general constant when I varies over the pattern m ~ . Unless we invoke the condition assumed in case I it does not make sense to speak of the conditional distribution of I over m. The situation can be saved though.
-
105
-
Theorem 6.~. If the probability distribution P of the deformations i~ covariant with respect to G, the probability P(E ~ I m % ) is well defined for any G-invariant set E ~ optimum problem is correctly formulated. Write P ( E ~ m )
S~, so that the
in the form
~ p(I# I~) H(dI~) and introduce the subse~m 81,$2,...,S@ and S o of ~
I s~ = {I~l Q(m~) p(121m@ > Q(m~)p(I~l m~), all ~ ~
} (6.5)
So
= [Iml Q(m~)p (I ~ # m~) = Q(m~)p(l & I m~)
for some
If ~ (S o) = 0 then partition into S? gives us the best recognition procedure among all G-invariant ones. Proof:
We have, since P is covariant, p (~l
and since ~
gi) = P ( ~ i ~ )
so that the function P ( ~
sure ~
: p ( ~ g-1~),
(6.6)
is invariant with respect to the synonyme transformations it follows that
P(~Ig~)=P(~)
bability
= P(g~)
=PC~r~),
(6.7)
I) is constant over patterns. Now a finite number of pro-
measures can always be written as integrals with respect to some fixed meaand with integrands, say p(l~J m), which are essentially
Radon-Nikodym deri-
vatives. Here we shall consider P(E ~ | m) as defined on the ~ -algebra obtained by restriction of the original ~ -algebra to G-invariant sets. We how have to choose G-invarlant subsets EI,E2,...,E ~ of U ~ a n d use them to discriminate between the underlying pure patterns, by deciding on m r if the deformed image ~ e
E r. We want to maximize the
total probability of correct decision P(correct decision) =
f r=-1~ Q(mr) J r
P(I~mr)~(d~).
(6.8)
But now we have reduced the problem to a form where the Neyman-Pearson lemma can be applied and this proves the statement of the the@tern, if we choose E r = Sr, r=-1,2,..,~, - If the set S O happens to have positive probability the result can be modified by the
-
106
-
usual randomization argument. So far it is clear, at least in principle, to decide on what recognition procedure to choose. But what do we do if
P(E~Im) is not defined but instead P(E ~ I I) varies when
I runs through a pattern m? Or when we are in case 3 and have no probability distribution at all over images or patterns? Our choise must then be based on criteria whose naturalness
is not quite as convincing as in the discussion above. If the
given but P ( E ~ I m )
Q(mr) are
is not uniquely defined we could think of a Bayesian approach by
postulating an a priori distribution over the (pure) images contained in each pattern. The reader may think of using the a priori probabilities belonging to an invariant measure with respect to the group of synonyme transformations.
Or one may use a minimax
approach. This is to some extent a matter of opinion. Personally the author would fav o r an appeal to the likelihood ratio principle,
this attractive and intuitively mean-
ingful idea that supports so many of the standard statistical decision functions. What has been said here will be illustrated by two simple but typical pattern recognitions. Theorem 6.4
Consider the graph patterns of Theorem 4.2. The deformations d and their
probability distributions are defined as Iij d:
Iij
>
(I
with probability I - p
11
- Iij
"
p
all the segments, i < j, of the graphs are transformed in a (statistically) independent mr way. Select a prototype I from each pure pattern m. If a deformed image belonging to pattern m e d~ is presented,
recognize it as generated by that pure pattern mr g JC p
that makes the quantity (I ~ ,I
r(m r) Q(mr ) (V~_p)
~I~
mr
)
(I p.~) m
as large as possible. Here we have introduced the inner product (I',I'')
=
~
i,j
(I'
ij
, I'').
ij
(6.9)
-
107
-
This decision procedure has the largest possible probability of correct recognition among all G-invariant recognition functions. Proof:
The deformations are not covariant with respectto the synonyme transformations
as can be seen from examples. On the other
hand, for any g 6 G, the transformations
dg and gd have the same distribution so that the probability distribution P of the deformations is covariant with respect to G. We can then find the best (G)-invariant recognition procedure by applying Theorem 6.3. To compute P(E ~ I m) we choose a prototype I m from each pattern m. For any G-invariant subset E a C ~ ~ w e p(E ~} Imr) = ~ -
P ( I ~ I I mr) =
~
p~(I~' Imr) (l-p) ~
z~
I~E E ~
get -I
(6.10)
E•
where ~(I', I'') denotes the number of pairs (i,J), with i < j ,
for which l'..~l~ l!~.zj
But (I',
I,,)
= ~
i<S
(IiS-z~)
~
2
=
~-
l• < a •
z,
+ 2~ lij ,,i< j
l~
(I', i '')
(6.11)
with the inner product (z',I")
=
>'- ( I ' , Z " )
(6.12)
i,j ' = I'' =0. Hence ~ can be expressed through the invariants r(I') and the convention Iii and r(I'') and the inner product -j ( z , , z , , )
= r(Z')
(I',I''), see Section 4,
+ r(Z")
-
(6.13)
(I',Z")
so that
r ( I ~ ) - ( ~ , I mr) P(E ~ I I m) = (1 -p)
(1-~_p)r(mr)
(6.1#)
where the notation r(m) is justified since r is constant over any pattern. The frequency function we should consider (with respect to counting measure) is then proportional to r(m r) + r(m)
(i~imr) I~
m
Having observed a deformed ~mAge belonging assign it to that of the pure patterns
to the deformed pattern m ~ ~ we should
ml,m2,...,m ~ that makes
108
-
I@~
-
m
as large as possible. Exa~le
6.~
See section 4 for notation. For n = 4, with ~ ding to the patterns with belongs to m with
~
~=
32 and ~ =
p = (~,m2)
with m I and m 2 correspon-
62 suppose that the observed deformed image
= 33. Let Q(32) = Q ( 6 2 )
= 1/2. We find r(32) = 1, r(62) = 5,
r(33) = 2 and the inner products.
(12, 32) = 0 (18,
]
(12, 62) = 4
32) = 0
(18,
(33, 32) m 2
62) = 4
(33, 62) = 2
so that we get the opt~msl recognition rule
'If p < 1 / 2
choose the pure pattern ~6 = 32
If p > 1 / 2
"
If p= 1/2
"
"
"
~ = 62
randomize with equal probabilities.
To illustrate an approach when the deformations ~ distribution we turn to the following p r ~ l e m
not have a covariamt probability
which is capable of considerable genera-
lization. Given a square lattice F = { (i,j); j = - L , - L + I , . . . , L - 1 , L } images Iij,Iij,...,I
of the plane and
defined for all i,j and vanishing outside the frame F. In
actual applications the values of the images will typically vanish everywhere except in a small part of F, but this nee~ not concern us here. The synonyme transformations g will be the translations operation on our contrast images
gl = {
(6.15)
i÷hJ÷kJ
so that the g's identified with p ~ r s
of integers and be written as g = (h,k). The de-
formations d will transform the pure images a c c o r ~ n g =
~
lij + xij
for ( i , ~ ) E
1
0
otherwise
to the rule F
(6.16)
-
109
Note that this is made of a mask transformation,
-
see earlier part of this section, and
additive noise. To fix ideas let the distribution P be given by the rule that all the x's are independent N(O,@).The quantities dgI and gdl do not even have the same range so that P can be covariant only trivially. Some possible recognition procedures are given in l~neorem 6.5. Theorem 6.~ a) With the given set up likelihood ratio recognition procedure is given by the rule: recognize the pattern m r if the quantity infFI I gIII 2
_ 2(i ~ ,gI)J
takes its
smallest value for a pure image I in the pattern m r. b) With the same set up but modified by assuming all pure images to be periodic (period L), Ii+nL,j+n L = Iij, the recognition procedure is given by the rule: recognize pattern m r if the quantity sup (I ~ ,gI) takes its largest value for a pure 9 image in the pattern m r. This could be called recognition b,y maximum covariance. c) If the pure images are periodic L, if the symonyme transformations are defined by g x I
~ a Ii + h, j + k
+ b
with arbitrary integers h,k, arbitrary real b and arbitrary non zero a, the recognition procedure is given by the rule: recognize pattern m r if the quantity
sup J correlation coefficient between I ~ and
Ii+h,j+ k I takes its largest value for a pure image from the pattern m r. This could be called recognition b,y maximum correlation. Proof:
a) The deformed image dgI vanishes outside the frame F and the (2
L + I) 2
remaining signs that make up the image have the joint frequency function p(I~1 gI) = const- exp (-
I 4 2
II ~
-gI
II 2 )
(6.17)
with the usual quadratic norm. We should search for a group element g making this likelihood function large, or what is the same thing gE~
II I
-giIJ 2
=
+
F II
I112 _
,g )3
-
b)
110-
If the pure images have the period L, the norm
~Igll~ does not depend upon
the
synonyme transformation g so that we should search for the g that makes the inner pro, duct (I~,gI) as large as possible. I!
c)
In the third case we can first minimize III~ - glJ!
2
with respect to a,b, and get
the "residual variance". Variance (I ~ ) ~ I - correlation coefficient squared (I ~, I)] so that inf#!l - gl II 2 g~G is realized for the translation h and k making the correlation coefficient as large as possible in absolute value. - In this third case the synonyme transformations contain contrast changes,
7. Discussion.
see section 3.
By now we have some idea about the grammatical approach to the analy-
sis of patterns, albeit only under the restriction that the synonymity transformations form a group. Starting from a set S of signs, perhaps subdivided into different categories, we form configurations.
In the set of aam~ssible configurations relations are
introduced b y indentifying certain configurations which seem the same to the observer. O f t e n one is helped by the fact ~hat the ensuing s t r u c t u r e ~ matical concept w i t h known properties,
is an established mathe-
but we must also expect to meet new mathematical
structures for more complicated patterns. A group G of synomyme transformations defined on T w i l l
tell us what images have the same meanin~
to the observer. We get classes of
semantically equivalent images:patterns. We could call this a pure ~rAmmar
~
:
~
of patterns
and write it as
(S,~,~,G,~).
The analysis of a given image I will be made in two steps I)
analyze I into signs, templates,
similarity transformations,
2)
assign I to a pattern m, perhaps labelled by a prototype Im, and find the corresponding synonyme transformation that carries the prototype into the given image.
To actually carry through this analysis we may use standardization of images, probes eto. In the practically more important case of disturbed w i t h a pur grsmm~r of patterns
~
images we start in the same w a y
, but now we must take into account what deformations
-
d a~that
111
-
can change the given pure pattern into the observed one. We then talk of a
deformation grammar ~ o f
patterns ann write it as
The first two steps of the analysis will look the same as for the l~ure grammar but can be based on completely different considerations I)
decompose I into elementary parts
2)
assign I to some (pure) pattern
3)
measure the proximity of this classification.
In the important case when we have probability distribution P governing the deformation we speak of a metric deformation grammar of patterns
In linquistic t e ~ s
and using the terminology of de Saussure it may be said that a pure
grammar describes the langue of patterns in contrast to a deformation grammar which deal8 with the parole. In this paper special patterns have been studied only as examples and then very briefly. The author will give elsewhere the result of detailed grammatical analyses of concrete patterns. It will also be shown how the above grammatical analyses has to be changed when the grammar is not given completely beforehand but has to be adjusted to fit the observed images, explorator~ ~rAmmars.
-
112
-
On the Potential Theory of Linear, Homogeneous Parabolic Partial Differential E~uations of Second Order. Siegfried Guber
I_~.
The following result is well known (~2S) :
Let Xs= R n+1 be (n+1)-dimensional euclidean space and for every open set U c X l e t ~ U be the set of all classical solutions of the heat conduction equation 2
nu-
~__a__u= o
(z~ s= ~
~xn+l If in addition
3~: U - ~
~
).
i=1
U for every open set U of X, then ( X , ~ )
is a harmonic space
in the sense of H.Bauer ([2J or L33). But (X,SF) is not a Brelot harmonic space
2._~.
( ~ 2S or [3J ).
I f 3 ~ U stands for the set of all classical solutions of A u
set U of a domain Y a R n , then ( Y , ~ )
= 0 in the open sub-
even is a Brelot harmonic space. Madame Herr@ has
Showm in her thesis ([ 5~) that (Y, S2) is also a Brelot harmonic space open U c Y
if for every
~2 U indicates the set of all classical solutions of Du = 0, where the op-
erators
i ,k=l
i=1
is defined in Y, and has locally lipschitzean coefficients with aik = aki such that the n quadratic form ~ aik(X) / i / k is positive definite for every x ~Y. This means i,k=l in particular that D is an elliptic operator. Madame Herr@ has extended her results in a series of three papers ( E6],ET~,~8~, ) on divergence type elliptic equations with merely measurable coefficients and where the solutions are to be understood in the sense of distributions. In the following paragraph we will sketch that within H.Bauer's theory there exists a result analogous to Herv@'s result stated in (ESJ).
-
~. ~U
Let X, = R n+l ( n ~ l )
113
and%~ :={ U c X :
-
U open} . For every U £ %~
we denote by
the set of all real functions u on U, which are twice partially continuously
differentiable
in xs = (Xl,..O,Xn) and continuously differentiable in t:= Xn+l, and
which are solutions of Lu = O; further let L ~ ,
U
~ L~ U
(U ~ ~
). The parabolic
operator L is given by n
2
n
, :
÷
i, k=l
bi(x,t
÷ o(x,t
i=1
where the coefficients ef L are defined on X and satisfy the following conditionss
(~)
( i,k~ {1,...,n 3 )~
alk = akl ~a
(#)aik; x~------~k, bi, c
(i,k,j & { 1 ,...,n } ) are locally H61der-continuous;
there exists a real A > 0 such that for every point (x,t) ~ X and every ~ | =
n
i,k=l for each to~ R exists a real
£ (t o ) =E >0, positive constants M,M1,M2,M 3 and
an integer m such thaw
I ai~Cx,t)l .< M I ( M + I ~ )
for a l l x ~ n
I b i (x,t) I < M 2 ( M + ~ )
"
c
Remark:
"
tl
~
, to + ~ tW
M 3.
Without any loss of generality we may
substitution v(x,t) Remarks
~
"
a n d t ~ 3 t o -~
assume that M3= O. This is shown by the
:= eM3tu(x,t). Therefore let c % O.
Regarding Herv~'s work, the first three conditions on the coefficients of L
look quite natural; not so condition ( ~ ) .
Probably ( 6 )
may be weakened;
but the glo-
bal character of axiom ~ in Bauer's theory ([3~) gives rise to the conjecture that there must be certain boundedness conditions on the coefficients of L. Addentum during the correction:
The boundedness of aik and b i is not necessary.
-
Theorem:
(x,L~) is a strongly harmonic
Sketch of the proof:(We solutions
~U
-
space in the sense of H. Bauer.
use the axioms and definitions
of Lu = 0 in U ~ t~
from the linearity
I~4
given in[3].) That the classical
form a linear subspace
of ~ ( U )
follows
immediately
of the operator L.
A x i o m 1:
That
L~:
U_+~
U (U~ ~ )
A x i o m 2:
We have to find a basis ~
only. Using the theory presented
is a sheaf is obvious. for the topology
of X consisting
sets
in[4] and [9] we can show that for every open cone
K = K(xo,te;r;t I) of X, having the open ball with centre base and the point
of L-regular
(Xo,to)gX and radius r > 0 as
(Xo,tl) , where t 1 > to, as top, the Dirichlet
This means for every f~ E ( K ~) there exists an Fe ~(K)
problem is solvable.
such that RestKF = | L ~
e
L~.
f We will call these cones standard
cones. Of course,
every standard
ely compact with non-empty boundary K ~, and the f a m i l y ~ sis for the topology
tially
at a point
of ~
we use the following
Let u be defined on U 6 ~
continuously
K is relativ-
of all standard cones is a ba-
of X.
To prowe the uniqueness Lemma 1:
cone
differentiable
(Xo,to)E U, then u(x,t)
, contimuously
result of L. Nirenberg
differentiable
([11]):
in t and twice par-
in x such that L u ~ 0. If u has a negative = U(Xo,to)
minimum
for every (x,t)a U with t ~ t o .
With the aid of this lemma we also obtain the result: f ~ ~(K~),
f~ 0
~
~K
~ o, which finally
shows that axiom 2 is fulfilled.
Axiom 3: We have to prove that for everyincreasing (h n) of classical
solutions
of Lu = 0 on U a ~ / ,
(x,t) of a dense subset of U, the function
sequence
(X has a countable base)
such that
sup hn(x,t) ( + ~ for all n sup h n is itself a classical solution of n
Lu = 0 on U. This fact is deduced, the non-negative Lemma 2:
in the usual manner,
solutions
of parabolic
(Harnack inequality)
that t < t o for every point for every u a
L ~ U , u>~0
from a Harnack
differential
Let E be a
standard
inequality
equations cone
which holds for
of the type Lu =0:
of U ~ 2 a n d
(Xo,to)~U
( x , t ) ~ E. Then there exists a real c o n s t a n t l y 0 :
such
such that
-
sup u ( x , t )
(x,t)~
Lemm~ 2 is a consequence
•~
115
-
yu(x°'t°)
"
of a result of A.D.Aronson
(~I~), which he kindly wrote to me
in a letter. This result of Aronson gives a generalization
of a Harnack inequality
proved by J.Moser in ([10~). Remark:
The elements of
L-harmonic measure,
L~ U (
U~
L-superharmonic
) are called L-harmonic functions.
function,
The notions
L-potential and L-absorption
set are
clear from ~2] and ~ 3 ~ •
Axiom ~ :
We have to prove that ( x , L ~ )
is a strongly harmonic
have to check the following two stements: (x2,t 2) a X, then there exist
space. That means we
(a) Given two diflerent points
~on-negative L-superharmonic
(xl,tl) ,
functions r,s on X such
that r(xl,t 1) s(x2,t 2)
~
r(x2,t 2) S(Xl,tl).
b) For every relatively compact set U ~ e x i s t s
a strict positive L-harmonic function
on U. First of all one can demonstrate Lemma ~ :
Let U~q~ , u defined on U and continuously differentiable
partially continuously
differentiable
in x such that Lu ~
in t and twice
0 on U. Then u is L-super-
harmonic on U. An immediate consequence
of lemma 3 and the assumption c ~ 0, is the fact that the con-
stant function 1 is L-superharmonic.
Therefore
valent with the existence of a non-negative
(take r=l in (a)) condition (a) is equi
L-superharmonic
function s separating
(xl,t I) and (x2,t2). The construction of such a L-superharmonic rather trivial in the case t I ~ t2; s(x,t) construct for an E ~ O 1 s u c h
function s on X is
: = e t is such a function.
that the coefficients
aik,b i ( i , k ~
If t I = t2, we
1,...,n~
dition (~) for this 6, a continuous function ~ in the strip ] t I -g ,t I + ~ Rest~t1_ ~ ,tl+&[~ is L-superharmonic
in the open strip It I - ~
satisfy consuch that
,t I + & ~ and separates
the two given points. This function ~ can easily be continued to a function u, which is defined and L-superharmonic
on the whole space X:
-
I 0 u(x,t)
, = l
We utilize
Lemma 4 :
, t ~t I - &
G(x,t) A
sup
116-
u
, t~t I -~, ,
t > t I
tI +8]
+
the following fact, which is proved by using again lemma I s
Every closed half-space of the form ~(x,t)st ~ to~ , with t o a R
fixed, is
a
L-absorption set, Statement (b) is obvious because there exists to every relatively compact U a ~
a stam-
dare cone containing U; the rest is, once more using lemma I, standard.
4.
(I) With the methods developed in E3S we can show that every point of X is L-polar.
From the general theory (~2S) we get the following two results: (II) Let U be a relatively compact subset of X. Then every point (Xo,to)~ U ~ such that there exists an open interval
S (xl,tl) , (x2,to)~ contained in U, where XlK XO~ x 2
and t I ~ to, is an L-irregular boundary point of U. (III) If U is a relatively compact subset of X, then every point (Xo,t o) g U ~ such that U c ~ (x,t) ~ Xs
t~t o~,
is an L-regular boundary point of U.
-
117-
Bibliograohy [13
Aronson,D.G.
Private communication
[2)
Bauer, Heinz
Harmonische R ~ u m e u n d
ihre Potentialtheorie,
Lecture Notes,to be published by Springe~-¥erlag Berlin-GSttingen-Heidelberg. [3]
Bauer, Heinz
Recent Developments in Axiomatic Potential Theory, published in this volume.
[4]
Friedman, Avner
Partial Differential Equations of Parabolic Type, Prentice Hall, Inc. 196@
[5]
Herr@, R.M.
Recherches axiomatiques sur la th@orie des fonctions surharmoniques et du potentiel, Ann. Inst.Fourier, 12
[6]
Herr6, R.M.
(1962)
415 - 571
Un principe du maximum p o u r l e s
sous-solutions locales
d'une ~quation uniform@ment @lliptique de la forme Lu
=
~
(
%
Ann.Inst. Fourier, 14
[7•
Herr6, R.M.
(1964),
Quelques propri@t~s des fonctions surharmoniques assocites ~ u n e Lu . . . .
@qua~ionuniform6ment
Hervg, R.M.
(1965),
215 - 223.
Quelques propri@t@s des sursolutions et sursolutions locales d'une @quationuniform@ment Lu . . . .
@lliptique de la forme
vergleiche [6]
Ann. Inst. Fourier, 16 [9]
@lliptique de la forme
vergleiche [6J
Ann. Inst. Fourier, 15 [8]
493 - 507.
(1966),
Ilin,A.M. -
Second Order Linear Equations of Parabolic Type,
Kalashnikov, A.S. -
Russian Mathematical Surveys,
Oleinik, O.A.
17,
[10] Moser, J.
(1962),
1 - 143
A Harnack Inequality for Parabolic Differential Equations, Comm.pure appl.Math., 17
11] Nirenberg, L.
(1964),
101 - 134
A Strong Maximum Principle for Parabolic Equations, Comm.pure.appl.Math.,
6
(1953),
167 - 177
-
Invariant
1 1 8 -
and j non-invariant measures
Konrad Jacobs
I.
Introduction
The purpose
of these lectures is to give a survey of some work on invariant
and non-
invariant finite measure in the ergodic theory of Polish space, w h i c h I did in the last two years. A generalization
of Poincar6's Recurrence Theorem from finite invariant to
finite
is given. A systematic
'recurrent' measures
shows that they often are
'wandering'
investigation
of recurrent measures
i.e. have mutually orthogonal transformed measu-
res; w a n d e r i n g measures do not a~m~t a stronger finite invariant measure,
hence the new
t h e o r e m in general cannot be deduced from the classical Poincar~ Theorem b y passing to such an invariant measure. Among the recurrent measures the subclass of special interest.
'weakly almost periodic'
If the underlying Polish space is compact,
measures is of
every w e a k l y almost perio-
dic measure gives rise to an invariant measure b y almost periodic averaging. to k n o w w h e t h e r this averaged measure is ergodic,and,
more generally,
We want
what its spectral
type looks like. U n d e r the assumption that the original w e a k l y almost periodic measure is
'uniformly mixing', we can prove directly that the averaged measure is ergddic.
is, however,
not w e a k l y mixing in general;
are different from I; to be precise, generated b y all e i ~ original measure.
there are
It
eigenvalues in its spectrum which
its point spectrum is the multiplicative
group
which occur in the spectrum of the almost p e r i o d i c i t y of the
In order to prove this result, we interprete the averaging process as
a channel from some Behr compactification
of the integers into the given space;
the ave-
raged measure is then the output source of this channel, p r o v i d e d we take the H a a r measure as input source. states that a 'mixing'
The desired result follows then from a general theorem which
ch~el
In our case, the channel
does not enlarge the poiDW spectrum of an input source.
exactly preserves
the spectrum of the H a a r measure which we
take as an input source. I n the last section I propose measures.
an individual
ergodic theorem for weakly almost periodic
It is shown that it cannot be true if we deal with averages
of arbitrary
119-
-
bounded measurable nuous functions. but at present
functions,
and that it is reasonable
Width this restriction
in detail
will be published
II. An extension
The result of chapters
in the Proceedings
II and III are going to be
of the Fifth Berkeley Symposium.
Those of IV
elsewhere.
of Poincar@'s
1. The classical Poincar~ Let
the theorem is actually true in a special case,
I cannot prove it under fairly general assumptions.
The present paper is rather sketchy. published
to restrict oneself to conti-
Recurrenoe
Theorem
theorem.
( ~ ,B,m,T) be a dynamical
system with finite T-invariant
mT = m( i.e. m(GT -1) = m(G) for every G ~ B G
=
~
)). For every U
measure
m (~)
=
I
B let
U T -t
t ~ 0 be the set of all points o f ~
which every meet U (on walking forward).
G T -1 =
~
U T -t
t~ is the set of all points i n ~ G
-
c
G
1
which ever meet U after at least one step. The set
G T -1
is the set of all points of U which never return to U. The set has (by the invarianee and finiteness
of m) the same finite measure
If we define Ure t c_ U by
as its subset GT -1, hence m(G - GT-I)=O.
U - Ure t = G - GT -I , we have finished the classical
proof
of the classical Theorem I
(Poincar@'s
Recurrence
Let m be finite and invariant.
Theorem~ z
Almost every point of any given set U a B will return to
U: m (Uret)
=
m (U).
There is a second proof which will yield more. Let t > 0 be fixed.
For every u with 0 ~
u ~ t consider the set
- 120 -
Uc
Un
T-ln
( U c denotes the complement/~
... n U c - U of U )
T -u of all points of U which are not back to U
until time u. B y suitable transformation the sets U Un
U c T -1
U~U
c T-ln
U
T-2
@erie
U ~ U ~ T-qn
Uc T-2c
• .. n U c T -(t-l)
U~
Uc T - 2 n
• ..
U c T-In
n
Uc
T -(t-1
)
o U ~ T -t
pass into the sets U T -t U
U T- ( t - 2 )
n
T -(t-1 )n
U c T- ( t - 1 ) o
U c T -t U c T- t @ o o o e o
I;T-I n
U c T-2n
...n
U c T-(t-2)N
U n U= T - l n
Uc T - 2 ~
...
U~
n
T-(t-2)~
U c T-(t-1)n
U c T -t
U c T -(t-l)
U c T -t
n
w h i c h are imutually disjoint, and actually form a disjoint decomposition of the set U
T -u
o~
kJ
o~u~.t
uT -u)
= m(U) + m(U~
U C T-I)
+ m(U ~ U ¢ T - I ~
Uc T -2)
oooeo
+ m(U~
U ¢ T-I~
U ¢ T -2,~
... ~ U c T-(t-1)~
U c T-t).
H e r e we have a partial sum of series whose members are decreasing and non-negative. A s the partial sum remains below 1, we conclude that =(U - U r e t )
: m(Un Uc T-ln
Uc T - 2 n . . . )
-
=
lira m ( U n t -- 05
U c T-Io
~21
-
... n U c T -t)
thus obtaining a second proof of Poincar~'s
theorem. We may, by the way, continue the
above computation by m(
~ U T -u) o .~u.~ t
= m(U n U T -1) + 2mCUn
u c T-lo
U
T -2)
+ 3mCUn
u c T-lo
U c T-ln
U
T -3)
. . o . .
+ Ct-1)mCU n U ~ T - l n
+ t m C U ~ U ~T - 1 0
...
n U ~ T-Ct-2)n
... n U C T - ( t - 2 ) ~
U
T-Ct-1))
U ¢ T -(t-1)n
U C T-t)
passing to the limit for t-* oo and denoting by r ( ~ ) the time of the first return to U of the point co~ U 5 we obtain Theorem 2 (Kac.) Let m be invariant and m(gZ) = 1.Then for every U e B
uT-u . o4 u~_ t If ( ~
,B,m,T) is ergodic and re(U)> 1
The latter
statement
is
frdm
very
0, then the right member is 1, hence, =
suggestivel
1
a point
oDE U n e e d s
a lozg
time r in
the
average
at le&B~5 to find back to U if m(U) is small. We don't use Kac's theorem later, but the reader should remind the second proof of theorem 1 which we obtained on our way to theorem 2. This second proof may be compared with the first one i~n a similar way as Euler's proof for the infinity of primes with Euclid's ancient proof.
-
122
-
2. The topologized version of Poincar~'s Recurrence Theorem From now onward we shall assume t h a t ~
is a Polish space (i.e. a separable topological
space which allows a complete metric) and T is a topological automorphism o f ~ (i.e. continuous as well as its inverse. B is now the ~ -field generated by the topologY. A point G o c ~
is called recurrent,
that
if there is a sequence O ~ tl~ t 2 ~ ...of integers such
tk oD T
as k -~ oo. Let ~ r e m
~
co
denote the set of all recurrent points.Let S = ~U 1,U 2,...
]be any
countable basis of the topology. From
62 rec
=
~O=
Ure t
u (~ -U ~)
and theorem 1 we deduce
Theorem ~.
Let m be finite and invariant. Then m-almost every point is recurrent:
m (~rec)
=
m (~).
This topologized version of theorem 1 is closer to Poincar~'s original ideas than theorem 1 itself, w h i c h is, as a matter of fact, obtained from Poincar~'s ideas by distilling away everything which is not pure measure theoz~. It is theorem 3, which e.g. implies that in the decadic expansion of almost every real from ~0, 1> every finite sequence of digits occurs infinitely often if it occurs at all, and it is theorem 3 which we are going to generalize now.
2. The extended Recurrence Theorem
For finite measures mk, m in the Polish s p a c e ~ fined b y
the weak convergence
mk--~ m is de-
jf dm k
>
am
(f bounded and continuous).
Let us now call a finite measure m in ~ recurrent if there is a sequence O w t 1 ~ t 2 < w t 3 ... of integers such that m T
tk
----> m.
-
123
-
We shall now prove the Theorem 4 (Generalized Recurrence Theorem). If the finite measure m is recurrent, then m-almost every point of
m ( ~ r e c)
=
is recurrento
m (~).
The simplest proof of this theorem is due to V. Strassen and goes as the first proof of theorem 1. One may restrict one's attention to some member U of some countable basis of the topology. It is sufficient to prove that for G = tVo U T -t
the decreasing se-
quence G ~ G T-I~ ... of open sets is not really decreasing in measure if m is recurrent. Now
m(G T -tk)
and by m T t k ~
=
(m T tk) (G)
m and a well knowm criterion for weak convergence this decreasing se-
quence has a limit inferior not smaller than m(G). Hence the sequence m( G T -t) is constant, and especially m (G T -1) = m(G), which proves m (Ure t) = m(U). But also the second (Kac's) proof of the theorem 1 may be generalized and yields a proof of theorem ~. One has again to pass to even a subsequence of the tk's , and to use a oasis of sets U which are 'm-boundaryless'
as well as all their transformed sets.
The reader should observe that theorem 4 states a hereditar~ property of recurrence.
III. The structure of recurrent measures 1. Examples A two-sided bounded sequence (xt) t integer of reals is called recurrent if it is a recurrent point in the space of all bounded real sequences (to which shift is applied), i.e. if there is 0 4 t 1 ~ t 2 < t 3 ~ . . .
such that xt+tk --~ x t (t integer)
for k - ~ c o .
It is easy to construct such sequences for given tl,t2,.., provided that tk+ 1 - tk-~co : Choose an arbitrary x o and put Xtk = x ° ( k=1,2,... and put Xtk_l = X-l'Xt k +1 an x t twice, but tk+ 1
(k=1,2,...)
) . Next choose x 1,x I arbitrary
and so on. One has to be careful to not define
- t k --~ co implies that any step this difficulty arises only for
a finite number of indices k, and for these we overcome it by considering the first definition as definitive.
- Another way of finding recurrent sequences rests on the
-
124
-
observation that every almost periodic sequence (xt) is recurrent. E.g. x t = cos oCt (t integer)
defines a recurrent sequence, however the real ~ i s
chosen.
It is clear that we may define recurrent or almost periodic sequences which remain between arbitrarily given real bounds, periodic =
and this enables us to define recurrent or almost
sequences (pt) of probability
distributions
( ~ ) k ~ A in a given finite set A =
~l,...,a} . The product measures m =I ~
= (''''~-1" ~o'~1
=T~ t
pt
in the corresponding
'''')I o t = 1,...,a (t integer) ~
examples of recurrent measures
represent the simplest
(with respect to the shift T i n ~ A ) .
to define recurrent measmres which are M a r k o v i a n m e a s u r e s
shift space
It is also possible
in,etc.
Being able to dispose of a reasonable variety of concrete recurrent measures,
the que-
stion arises what can be aaid about the structure of the system of all finite recurrent measures in a given Polish s p a c e ~
with a given automorphism T.
Easy examples show that the sum of two recurrent measures m I and m 2 need not be recurrent; we need only to restrict our attention to a situation where (mI + m2) T~k--*m1+m2 always implies
-~m2, and then construct, within this situation, tk tk two measures m 1,m 2 which cannot fulfil m I T ~ m 1,m 2 T --~ m 2 with the same sequence
tl,t2, . . . . .
m I T~--* m 1 and m 2 T k
Such a situatio$ is e.g. given if m 1 and m 2 are point masses of unequal
size im ~he above shift space a2
, and it is not hard to construct m 1 and m 2 as re-
quired. Thus we are lead to a modification
of our intentionss We shall fix a sequence
O ~ tl< t2< ... of integers once for all and consider only those measures m which fulfil m Ttk--~ m for Just this sequence. 2. The c~ne R + :(:$~2j.o.~ Let~be
and the simplex V (~a,t2..o.)
a polish space and T an automorphism o f ~
fixed sequence of integers, R+(tl,t2,...)
, moreover let O W t 1~ t 2 ~ ... be a
and denote by
the set of all non-negative measures
in
such that m Ttk--~ m
( including m = 0 ) V (tl,t2,...)
the set of all probability m e & I ~ e s
in
~+ (t 1,t 2,..-).
-
125
-
It is nearly trivial that R+(tl,t2,...) is always a convex co'no with vertex 0, and that V(tl,t2,...) is a convex set, and that T leaves each of these sets invarianto Easy examples show. however, that V(tl,t2,...) is not weakly compact in general. If is the shift space, and t k = kl, then V(tl,t2,... ) contains all periodic normalized measures, and these form a weakly dense subset of the set V of all probability measures in that space; but it is easy to construct some m such that m T k__~ m does not hold. Hence V(tl,t2,...)
~
V, although it is dense, and consequently V(tl,t2,...) is not
weakly compact. We cam, however, prove Theorem 3The convex cone R+(tl,t2 ' ...) is a lattice, and V(tl,t2,...) is a simplex. For the proof, we consider the space R(6Z) of all finite signed measures h in ~1 , endowe it with the weak topology and define the concept of a recurrent signed measure in the obvious way. Now denote by R(tl,t2,...) be the set of ~ll signed measures h i n ' s u c h that h T tk--~ n . Clearly R(tl,t2,...) is a linear space. In order to show that it is a lattice, it is sufficient to show that it is closed under the lattice operation h - ~ ~hl. Now for every bounded continuous function f ~ 0 on L (h)
= ~
dthl
, the functional
(h signed measure)
is a lower semicontinuous function on the space R(6~ ), with respect to weak topology. This follows from the fact that is the supremum of all those functions on R ( ~ ), which are of the form
where fl' .... ,fn.> 0 are continuous and ~- fk = f' ~ e k tions Lfl .... 'fn are weakly continuous on R(~Cd).
from the fact that these func-
Taking into account the general relation lhlT t >~ Jh Ttl, we obtain llm inf J f k
dlhlT tk ~> lira inf J f dlh T tk' = lira inf L(h T tk) ~> L(h) = J f dlhl, k k
-
126
-
since L(h) is lower semicontinuous. Applying the same inequality to (const) - f, where const~ f, we find equality, hence
lh IT
lhl •
Now we know that R(tl, t2,...) is a lattice. This implies theorem 5. As V(tl, t2,...) is not compact in general, we cannot assert generally that it contains any extremal points. However, we have the Theorem 6.
If a measure m (1)
lim
V(tl, t2,...) is mixing, i. e.
[ ~ f T t gdm
- J f T t dm
fg
dm
~
if
=0
for arbitrary bounded contimuous f, g, then m is an extremal point of V(tl,t2,...).
Proofz It is easily seen that the validity of (1) may be extended to arbitrary g 6 L ~
(this
is due to the fact that T doesn't act on g in (1); for f such an extension is impossible in general, as we shall see). For the extremality of m in V(tl, t2,...) it is sufficient that every m'~ R+(tl , t2,...) which is absolutely continuous with respect to m, is a constant multiple of m. Assume dm' = gdm; then we obtain, for every bounded continuous
dm' = 1
dm'T
=llim j f T tk d m J J g L k i.e. m' = ~ m
with
~ =Jg
am
=
lim k
dm'
= ~lim ff dm T t k ~ J g ~k V dm = m'(A~),
= lira k dm
dm = Jf
=
dm ~ g dm
q.e.d.
Theorem 6 says that a mixing measure is extremal in every V(t 1, t2,...) to which it belongs. Since the recurrent product measures in shift space constructed in section I are evidently mixing, we can dispose of some extremal points.
Theorem 7 •
The automorphism T o f ~
permutes the ex~remal points of V(tl,t2,...)
(if there are any). The proof is trivial. Theorem 5 implies
-
127
-
Theorem 8 . The extremal points of V(tl,t2,...) Call a measure
(if there are any) are mutually orthogonal.
m
invariant, if m T
=
m
periodic with period d > O, if m T d
=
m
wandering,lf the measure m T t (t integer) are mutually orthogonal: m Tt/~ m T s
=
O
(t
~
s).
As a consequence of the preceding theorems we obtain Theorem ~ . E v e r y recurrent measure which is an extremal of some V(tl,t2,...)
is
either invariant or periodic or wandering Especially the non-periodic ones among our recurrent product measures shift space are wandering. F o r a wandering meamure m it is a apparent that (1) cannot hold for arbitrary bounded measurable f, g because it is easy to find a strictly invariant set F such that m Tt(F) = I/2 (t integer), and f = 1F, g = 1 ~ _ F would not fulfil (1). It is also possible to find a set F that m(F) = m(/&)
and F T - S n F T -t = 0 (s ~ t), hence Fre t = O,
w h i c h shows that a direct generalization of theorem 1 to recurrent measures would be impossible,
and that we really have to choose the topologized version (theorem3) as
the starting point for our extensions given in chapter II.
IV. Invariant averages of almost periodic measures. F r o m now onward we shall only deal with almost periodic measures,
i.e. with measures m
f o r which the sequence j f T t am
= Jf
dm T t
(t integer)
is an almost periodic function on the additive group Z of all integers, for every boun~ ded continuous function f. Moreover we shall a s s u m e ~
to be compact. Denoting almost
-
128.-
periodic averaging by a bar, the Riesz representation theorem immediately yields a measure ~ such that
jf
d~
=
jf dm T t
.
the translation invariance of the almost periodic mean value implies the T-invariance of m:
m T
=
m.
m
is called the average of m.
It is our aim to investigate the properties of ~ in terms of the classical ergodic theory of invariant measures. Obviously we hope to deduce properties of ~ from properties of m, but the measure theoretical relation between m and ~ is very loose in general. If m is wandering, then ~ and m are orthogonal. However, it can be shown directly that ~ is er~odic if m is uniformly we call a system of probability distributions in
uniforml~ ~ n ~ ,
continuous f, g there exists reals d ~ > 0 such that ~ / f Tt g dm
-~
T t dm / g
dm ' ~ t
~t~
mi~In~. Generally if for arbitrary
O and
(t = 0,1,...)
;
we mention in passing that the weak closure of a uniformly m i ~ ng system is a again uniformly mixing. Now, a single probability distribution m is called uniformly miwin~ if the system of all its transformed measures m T t (t integer) is
uniformly mixing.
The almost periodic product measures in shift space constructed in III. I are uniformly mixing. Theorem 10 . If m is uniformly mixing then ~ is ergodic. We only sketch the proof:
For arbitrary
continuous f, g we have
~o
n-1
t-1
~o
~o
t-1
n-1
•~ o
"k---o
The approximation symbols ~ used here have to be made precise im terms of some a > 0 for n and t sufficiently large, l~ne middle ~ to almost periodic averaging. The relation
is due to uniform mixing, the other two
-
lim
fT
t
°
129
-
a~
=
11=0
is sufficient for the ergodicity of ~. E r g o d i c i t y is only the most primitive among the so-called spectral properties of am invariamt measure. Let us n o w investigate these properties of m more carefully. Generally,
if ~ is a T-invariant measure, then T maps the complex Hilbert space
into itself and in fact defines a unitary transformation T in L 2 . The spectral properties of this unitary transformation are also called spectral properties of m . Ergodicity is a spectral property= it means that all eigenvalues are simple. There are two simple examples, where the spectral properties of the average ~ of an almost periodic measure m are easily discovered.
Example I.
Let ~
= ~1,...,n~
and T a cyclic permutation o f ~
mass one at point I. Then ~ is the equidistribution i n ~
. Let m be the point
, and L 2 = R n, T permuting an
orthonormal basis cyclically. Hence the spectrum of T is the group of all n-th roots of unity. Example 2. and T = z
Let ~
= { z I z complex,
] z(
z ei~ the roSation o f ~ b y
then m is almost periodic and not periodic,
oc = I} be the unit circle line, ~ irrational the a n g l e ~
. If m = ~ f o r
some point o ~ ,
and ~ is the equidistribution o v e r ~
• The
unitary mapping T in L 2 has pure point spectrum, namely, the multiplicative group ~ ei~m
In integer~
generated by
e i~L
.
In both examples we see that the frequencies which occur in the 'almost periodicity of m' appear again in the spectrum of the unitary mapping T in L 2
•
We shall now see
m
that this is generally true. For this purpose we interprete ~ as the output of some stationary channel whose input has known spectral properties.A theorem which says that our chsnnel preserves the point spectrum will yield the desired result. Let us first define the concept of a channel: it is simply a stochastic kernel P(~',F) from a measurable s p a c e ~ ' t o
a measurable space h~. Any probability distribution m'
-
in 6~' is called an input for the chAn~el, is then called the corresponding
(F)
=/m'
N o w assume that measurable
(d~')
P (~',F)
automorphisms
=
o~' and F, then the c h ~ n e l
the complex Hilbert
T: ~ ' - - *
(fP)(~,) is in ~ ,
and P
(2)
=
P (~'
are given (we
, F) ,
P is called stationary.
A stationary
chRnnel P al-
output m. Moreover a channel
links
and L 2 : if f a L 2 , then m
contracts
fTP
the L 2 norm; the stationarity
(re
This gives us a chance to relate the point trum of T s L~,
d-A --~ ~
= f,(~,)
: f --~. f'
n~
GA' and T:
If
input m' into an invariant
spaces L 2
measure m = m' P in
output:
P ( ~ ' T -1, f T -1)
ways sends an invariant
-
and the transformed
choose the same letter T for both of them).
f o r all
130
~).
spectrum of T
• L 2 ,: If f ~ L 2 is an eigen-vector, m j
of P implies
: L 2 --*L 2
to the point spec-
i.e. if it fulfils
m
fT for some complex constant ~
=
~f
, then (2) implies that f'
f' T
=
The only problem is whether f
=
fP fulfils
~ f' ~
0. We are thus led to the investi-
0 implies f'
gation of the kernel K m
~
~
m'
Theorem 11 . If
~is
a polish space and T is a topological
che~nel P is mixiz~o b l e d in the channel
automorphism
i.e. if each of the probability is mixing,
then the kernel K of P
of~,
and if moreover the
distributions ! L2 , ~ L2 m'
P(oo',
.) assem-
is orthogonal
to all
-
131
-
eigenvectors of the unitary transformation T : L 2 m
--~L 2 .
We only sketch the proof. There are ~nown theorems on Hilbert 8pace which imply that it is sufficient to prove f Tt
r 0
(weakly in L 2 )
for every f a K as t-~ oo. Let (.,.) denote the scalar product in L 2
or L 2 . Then m"
f c K, g g L 2 implies m
(f
T t,
g)
= Jf
Tt g dm=
Jm'(d~') #PC~',d~)
-~jrm'(doJ') J ~ P ( ~ ' , =
( fP,
gP
)
d~)
=
f(~)
#
P(~',
f
Tt(cD) g(cO) d~)
--~
g(~)
o
The --~is due to m~wing combined with the invariance of m' and the stationarity of P and is immediately clear if f is continuous and bounded; for general f ~ K one has to add an approximation argument. This theorem tells us that for a mixing channel an eigenvector f ( ~ 0|) in L 2 is really i
sent into an eigenvector f' = f P
~
0 in L 2 , with the same eigenvalue. This m implies m'
Theorem 12 . If the stationary channel P is mixing, then the point spectrum of the output ~ is conrained in the point spectrum of the input m', with at most the same multiplicities. Especially if m' is ergodic
-
i.e. if all eigenvalues in L 2 m'
are simple - then
is ergodic. The same statements hold for some other spectral properties, namely weak resp. strong mixing. Let us now return to our original situationl ~ i s cal automorphism o ~
, man
a compact Polish space, T a topologi-
almost periodic normalized measure, and m
the normalized
invariant measure which is the average of m. For simplicity let us exclude the case where m is periodic. How can we find a stationary m~ving channel P and an input m' such that ~ is the corresponding output?
132
-
-
Let Z be the additive group of integers, and T = t -~ t + 1 Every probability distribution m i n .
the unit translation in Z.
Immediately defines a stationary c h ~ n e l Pc from
Z to ~2 by Pc(t, F)
=
(mT t) (F)
Indeed Po(t-
1, FT -1)
=
(mT t-l) (FT -1)
=
(aT t) (F)
shows that Pc is stationary. If m is almost periodic, then every continuous f on
6%
leads to an almost periodic fo = fP on Z = fo(t)
= JPo(t,
dec) f ( ~ )
=
J f dm T t
.
Let S o denote the system of all almost periodic functions on Z which we obtain in this way. It is well known that we may consider the fo ~ So as restrictions of continuous functions on a compact group~Z'~ Z, and that ~' is uniquely determined by S and the requirement that the continuous functions mentioned separate the points of e-A' . The compact group 6Z' is then called the Bohr compactification of Z with respect to the system S O of almost periodic functions. Let us now take just this Bohr compactificationgZ' as the input of a stationary channel P which is easily obtained from Pc by an extension procedure and is an assembly of almost periodic probability distributions,
actually the weak closure of ~ mT t I t in-
teger}. If we assume that m is uniformly mixing, then this weak closure is uniformly mixing tqo, and the channel P is a f o r t i o r i
mixing.
Let m' be the Haar measure in the compact g r o u p ~
. Since Haar integration is tanta-
mount to almost periodic averaging, we find
/ f d (m'P)
=
Jm'
(do~')
/P(~
', dto) f(oa)
= JP(t,doo)
f(oo) =
i.e. ~ is the output of P if we take m' as the input. Now we can apply theorem 12, and find ourselves left with the following problems only: (1) 2)
What is the point spectrum of m' ? Do some eigenvalues really disappear when we pass from m' to ~ ?
-
133
-
Concerning I) let us first remark that we take of course T : ~Z'--~d~' to be the translation by the generating element I of the subgroup Z o f ~ in~'.
'. It is known that Z is dense
This implies - together with the uniqueness of the Haar measure - that m' is
ergodic. Moreover it can be shown that m' has pure point spectrum, namely, the multiplicative group generated by the
e i~ which really occur in the Fourier expansions of the
fo ~ SO* Concerning
2) let us observe that the continuous functions f' c fP which we obtain for
continuous f are nothing but those continuous functions o n ~ Z ' whose restrictions constitute the system S o. They form a system S' which separates the points o f ~
'.
Taking into account the mixing properties of P and the almost peridicity of the f o e So we find that the uniform closure of S' is also multiplicatively closed, hence consist of all continuous functions o n e ' eigenvector in L 2 occurs in m' that of m'.
( Stone-Weierstrass
theorem ). This implies that every
L 2 P, hence the point spectrum of ~ is exactly equal to m
S1~mm~ng up w a obtain ( the periodic case may be included now again ).
Theorem 12 . L e t ~ Z be a compact Polish space dic under the automorphism T o f ~
and m a probability distribution which is almost perio. Then, if m is uniformly mixing, the average m has a
point spectrum which is simple and equal to ~he multiplicative group generated by the frequencies e i ~ w h i c h functions J f
Thus
actually occur in the Fourier expansions of the almost periodic
d m T t on the integers ( f continuous
). Especially,
m is ergodic.
we have reproved theorem 10 by a different method. Theorem 13 shows that m is not
weakly mixing (i.e. without eigenvalues ~ 1) if m is not stationary. It can be shown that ~ as a rule has not pure point spectrum. All these results apply to almost periodic product measures in shift space. They show, b y the way, how to obtain ergodic measures in shift space, which have a given group of eigenvalues.
-
134
-
V. A proposal for an individual er~odic theorem. Let T be an automorphism of the polish s p a c e ~
. For any finite-valued real function
f let us consider the averages t-1
(3)
1 ~
~u
f
( t = 1,2,...).
u=o Individual ergodic theory is interested in the convergence of such sequences in the sense of almost everywhere convergence with respect to a given measure m. Assume that is wandering. Then it is easy to find a bounded measurable f such that the sequence (3) converges m-almost nowhere. Namely, one can find a measurable set F such that the sets FT -t (t integer) are mutually disjoint, and m(F)
= m(~).
Now choose a
bounded sequence ao,alt.., of reals ~_ 0 such that the averages ~-1 t
u-=o
don't converge, and put
oo f
=
au 1 F T u . u~o
Clearly
t-1 ~ ~ f TU
=
~1
t-1 ~
au
m-a~most everywhere, which proves that f has the
U=O
U=O
claime~ undesirable property. This example shows that in our theory of recurrent measures m there is no reasonable theorem about the convergence of expressions (3) if we allow bounded measurable fumetions f, because wandering measures are a quite normal phenomenon in this theory. I propose therefore to allow bounded continuous functions only. Clearly convergence m-almost everywhere implies the convergence of the integrals t-1 (4-)
I
f
dm Tu
(t=1,2,...)
u=o
if f is bounded. If f is continuous, then we see that m cannot be allowed to be an a~bitrary recurrent measure; at least the sequences (4) have to converge for bounded
-
135
-
continuous f. Thus we come to the formulation of a Proposed Ergodic Theorem. Let m be a finite measure such that the behaviour of the sequences (4) for bounded measurable functions f is promising good averaging properties (e.g. let m be almost
periodic).
Then f o r
every bounded continuous f the sequence t-1
t
u~o
converges m-almost ever2where. It is easy to prove this theorem by means of Kolmogorov's law of large numbers, if m is a product measure in shift space. Indeed we can - by the Stone-Weierstrass-Theorem restrict ourselves to cylinder functions f in this case, and this reduction provides enough independence among the f,fT,fT2,.., in order to apply an obvious modification of the law. I don't know any proof of the above theorem in the general case.
-
136
-
Representation of Abstract L-Spaces
Demetrios A. Kappos
0.
An abstract L-space is a Banach lattice for which x > 0 ,
= llxll +[I Yll - Let ( ~ ,
y>0
implies
n x + y11
A, m) be a measure space, then the space L ( ~ , A, m) of all
real valued A-measurable and integrable functions defined o n , m o d u l o
functions vani-
shing almost everywhere (i.e. except on a set of measure zero) is an example of an abstract L-space, called a concrete L-space. S. Kakutani proved in [4~
: If an abstact
L-space B satifies the following property (K)
If
xAy
= 0, then II x + yll = llx - y
and possesses a weak unit e, then there is a probability space ( ~ ,
A, p), W h e r e 3 1 i s
a
totally disconnected compact topological space, A the B o o l e a n ~ -algebra (~ -field) of all Borel subsets of ~Z and p a probability on A, such that B is isometwic and lattice isomorphic to the concrete L-space L ( ~
, A, p).
Any separable abstract L-space possesses always a weak unit e and if it satisfies condition (K) then it can be embedded isometrically and lattice isomorphically into the concrete L-space L (2 ,A,p),where ( 9
, A,p) is the linear Lebesgue probability
space, i.e. the unit interval [ o, I J
on the real line R, A the B o o l e a n ~
(~ -field) of all Lebesgue measurable subsets of [ 0, I~
and p the
-algebra
Lebesgue measure
on A (cf. Kakutani ~4~ theorem 8). This theorem of Kakutani can be generalized in the following way (cf. Kappos [7~ ). Let ~ = (o~,
P~Ef ~,0P#)
be the cartesian product, where every factor
: 0~x
~I},
~he product probability space wlth A~ the B o o l e a n ~ -algebra of all
Lebesgue product measurable subsets o f ~
~)
Ef = ~ x ~ R
Any concrete L-space L ( ~ ,
and p~ the Lebesgue product measure on A~,
A, m) satisfies condition (K). If an abstract L-space
B does not satify this condition, then the norm of it can be provided with an equivalent norm namely
IIxll ~ = iIx+!l+ fix-If which satisfies (K) (of. Kakutani E 4J theorem I).
-
137
-
then for any abstract L-space B satisfying condition (K) and possessing a weak unit e, there is a smallest ordinal number ~ ~ 0 such that B can be belian lattice Isomorphically into the concrete L-space L ( ~ ,
embedded isometrically A,
p~).
We notice that an abstract L-space possesses first of all the algebraic structure of an abelianlattlce group, moreover of a vector lattice. These structures play the main role b y the representation theory. Namely, an abelian group G can under certain farther conditions be embedded into the space of the Carath~odory place functions generated by the Boolean ring G ~ formed of all carriers of G (cf. G o f f m a n ~ 3 ~
). If the algebraic struc-
ture of the Boolean ring G ~ is such that a strict positive probability p can be defined on G~then representation space of G can be the space of real valued random variables over the probability algebra (G~,p). However, a probability algebra can be represented b y certain probability space of the abovementioned kind (r~,, A~, p~) (cf. Kappos ~ 5 ~ and ~6S ), i.e. the elements of G can be represented by classes of real valued A ~ measurable functions defined o n ~ .
If G is an abstract L-space with a weak unit, then
the representation mpace is a subspace of a concrete L-space of the form L ( ~ 1.
Let G be an archimedian lattice group. ~)
Let moreover
, A~, p~).
P = G + be the positive cone
of G, then P is also a distributive lattice with a minimal element O. For every x ~ P, we define D(x) as the set of all elements in P which are disjoint with x, i.e. D(x) = =~y~P
: y~x
This relation
= O~ ,then we define the relation x ~ y
if and only if D(x) = D(y).
is an equivalence in P. Let G ~ be the set of all equivalence classes
x ~, y~, ... in P, then G ~ is a distributive lattice with minimal element 0 ~ = (0). We call the elements of G ~ the carriers of G (cf. Goffman ~ 3 U ). If G possesses a weak unit ~ )
e,then G X p o s s e s s e s
a maximal element, namely e ~ =
(e). If G is an archimedean
saturated (complete) ~s~) group, then G ~ is also saturated and moreover a relatively
I) A lattice group is called archimedeam if x ~ 0 and ny m x, n = 1, 2, ... imply y L O. We remark that any archimedeam lattice group is abelian and a distributive lattice. H) ~)
A positive element e ~ G is called a weak unit if e A x We mean here "conditionally saturated",
= 0 implies x = O.
i.e. every bounded in G set of elements possesses
infimum and supremum in G. We use here "saturated" instead o~'complete", for "complete" will be used later with respect to convergences
(cf. also Papangelou ~ 8 J and ~9~ ).
-
complemented distributive lattice,
138
-
i.e. a saturated Boolean ring. We remark that every
archimedean lattice group G can be embedded regularly and
isomorphically into a satu-
rated archimedean group ~ (cf. Papangelou ~9~ ). Hence the lattice G ~ of carriers of G can be embedded regularly and isomorphically into the saturated Boolean r i n g ~ E of carriers of ~. It is known if the lattice B ~ of carriers of a Banach lattice B is always a Boolean ring. However,
this fact is established if the Banach lattice B satis-
fied the following conditions (Goffman ~3~ ): 1. The norm is strictly increasing
: i.e. I xl>r~
implies
IIxll>llyll
2. E v e r y bounded monotone sequence in G is fundamental with respect to the norm convergence. It is easy to prove that an abstract L-space B satisfies conditions 1) and 2) so that its lattice B ~ of carriers is a saturated Boolean ring. In all cases in which the lattice G ~ of carriers of an abelian lattice group G is a saturated Boolean ring or it can be embedded regularly and isomorphically in a saturated Boolean r i n g w h i c h
is the
lattice of carriers of a regular extension of G, we can define the vector lattice of all Carath@odory place functions generated by G ~ and use ~ ( G ~)
(~(G~)
to represent the
group. In order to show this fact, we shall state briefly in the following Nr. a suitable process to define the vector lattice G (A) of all Carath@odory place functions generated by a saturated (or g -saturated) Boolean ring A. This process constitutes a convenient generalization of a constructive process which we applied (cf. Kappos E6J ) to define the vector lattice of all real valued random variables over a probability algebra (A, p), where however A is a saturated Boolean algebra, i.e. A possesses a unit e. 2.
Let A be a saturated (or ~ -saturated) Boolean ring. For any element a e A, with a~
@, let (al,a2,...) be a countable class of pairwise disjoint elements of A such that a = ~J
aj, further let fj,
fa
j~-
j aj
J ~ 1 be a sequence of real numbers and consider the form
~
aI +
a 2 + ...
as an elementary place function (briefly e. p. f.) fa over A. If the class (al, a2,...) is finite then fa is called a simple place function (briefly s. p. f.) over A. Let 1.a, a •@
~5~R, = e,
o.a V a ~[~R,
a A, be also considered as e. p. f. 's and put 1.a = a, o.a = e , ~ a@A.
-
139
-
The set of all e.p.f's over A will be denoted by E = E(A). The element 0 will be called the zero of E. Let and
fa = ~ l f j a j be two e.p.f's of E; we put
gb = ~- ~ ibi i>~1
a-b=bo ' b-a=ao' ~o =0 , ~ o =0 and consider the two e.p.
f's
o[jaj
and
aob = i>OZ ibi
then we define fa = gb if and only if, for every pair
(j,i)&
[ 0,1,2,...3x ~0,1,2,...~
with aj n b i { @ we have fj = 9i " Let r and @ be any relation and operation respectively defined in R, then the relation r can be induced in E, namelyz fargb in E if and only if, for every pair (j,i)&[ o,1,2,...) x [o,fl,2,...9
~
r?i holds in R. The operation @ fa ® gb
with a j o b i ~ @ the relation
can also be induced in E, as follows
= ~- (aj°bi)
(~j~i)
for all pairs (S,i)& { 0,1,2,...]×~0,1,2,...}
with a j ~ b i ~ ~.
In this way we can define the relations fa ~ gb' fa ~ gb etc. and the operations fa V gb' fa ~ gb' fa + gb 'fa'gb" A multiplication of a real number ~ and faeE can also be defined, namely: ~ fa = j ~ 1 ~ j a j
•
It is easy to see that fa = faub ' gb = gaub and that E is with respect to the opers~ions V, A , + amd ~ fa an archimedeam vector lattice in which a multiplication fa'gb is also called defined. If the Boolean ring A possesses a unit e, then the e.p.f. 1.a = a is equal to the indicator of a (cf. Kappos ~ 6 ~ ), i.e. a is equal to the e.p.f. I a = 1.a + 0.a c and the e.p.f. 1.e = e is a weak (Freudental) unit in E. The set I(A) of all e.p.f's of the form 1.a = a ~ a & A
is a sublattice of E, lattice
isomorphic to A. Let a be a fixed element in A different of ~, then the set R a of all e.p.f's of the form ~ a,~/~ ~ R is a regular and isomorphic to R vector sublattice of E. For any fixed element a & A, a { ~, let E(aA) be the set of all e.p.f's of the form fa = ~
~1 jaj
where a =
j~l[-~aJ
and al'a2'''"
pairwise disjoint, then E(aA) is a vector sublattice of E(A) in which a is a weak unit.
-
140
-
The so defined E(aA) is also isomorphic to the vector lattice of all e.p.f's over the saturated Boolean algebra aA = ~ x u A
, xga~.
An order convergence can be introduced in E(A). Obviously, E is not complete with respect to the order convergence. However , we can apply a completion process of Papangelou ~8] and [9] and extend E to a vector lattice ~
m E ~! which is o-complete, i.e.
every o-fundamental sequence o-converges in ~, and (conditionally) o- -saturated, i.e. co every bounded sequence fn e ~, n = 1,2,... possesses a supremum ~ fn and an infimum co -1 A fn in ~. The vector lattice E can be embedded into ~ isomorphically and g -regularly. n=l In order to define the extension ~,we consider the so called E-o-convergence ~). Let the vector lattice of all E-o-foundamental s e q u e n c e s
~Ebe
can be represented in ~ E
in E. The vector lattice E
by the constant sequences and is regular in ~ E "
Now let
E E = ~ be the quotient vector lattice of ~ E modulo, the lattice ideal of all B-o-null sequences in E, then ~ is o-complete and obviously E can be embedded 6--regularly and isomorphically in E. Let namely every element f a E be mapped in the class which is representod by the constant sequence fn = f' n = 1,2,...
. In order to prove that ~ is
-saturated, we have to prove (cf. Papangelou [9] proposition 4.7) that the following condition is satisfied in E z (P)
If the sequence fam~ E, n = 1,2,... is increasing and bounded in E, then there is
a decreasing sequence gn e E, n = 1,2,... with fam.~ gm' n = 1, 2,...and gm- fam~
~"
Let the sequence f e E , n = 1,2~...De oounded by an e.p.f, f a ~ E , i.e. fan~ fa' n = an = 1,2,... ; it can be assumed, without any loss of generality, that f ~ @ and J.A
a>~an,
n = 1,2,...
, then fa and fan are elements of E(aA). But E(aA) can be considered
as the vector lattice of all e.p.f. 's over the Boolean algebra aA.In this case condition (P) is satified in E (aA). (cf. Kappos [ 6 3 ) ; the vector lattice P is hence o-complete and ~ - s a t u r a t e d
and E can be considered as a regular vector sublattice of
~, which is dense with respect to the o-convergence in ~. We shall call W = ~(A) the vector lattice of all place functions generated by the Boolean ring A. It is easy to prove that ~.(A) is isomorphic to the vector lattice ~ ( A )
of
all Carath~odory place functions generated by A,as it is defined by C. Carath~odory [~] and
~2~ or Goffman [3] •
H) Regarding the notions in this Nr. cf.
Papangelou [8~ and [9] •
-
3.
141
-
Let G be any sawurated archlmedeam lattice group and G m = A the saC,,rated Buoleam
ring of all carriers of G, then there is a set e i E Gm , i a I , ments e~ such that for every x~ e
G ~ we have
of pairwise disjoint ele-
L ~ x ~ ~ e~ = x ~ : If possesses a weak
unit e then for the unit carrier em we have x ~ ral case Goffman [3] defined a collection ~
e~ = x
= { ei g G ,
for every x~g G ~. In the geneei ~
O, i ~ I }
such that for
H
every i e I the element e i is the carrier of e i as a generalised weak unit for G. Let ~ ( G x) = ~ (GI) be the vector lattice of all p.f.'s generated by G~, then we can define a mapping ~ of the positive cone G + = P of into ~(G x) which is one-one, order prese~,Img and operation preserving in the following way (GoffmamE3] Nr. 5)For every x g G + and (m,n,i) g N X
N
X I we put
f'm,n,i = and
V i~I
YXm,n=
Ym,n,i
NXN
(m,n) g
then
~m+l ,n >-" Y:m,n
m =
1
2,
and for every n = 1, 2,... we have
(of. G o f f m ~
[ 3] ).
If we wr~te ~,n
=
Y~m+ltn- ~m,n
m= 1 , 2 , . . .
V m=l
n=1,2,...
then =
n
where t h e e l e m e n t s ~ , n
' x~2,n ' ' " . pairwise disjoint.
Hence we c a n f o r m t h e e . p . f , s m
fxH,n = m ~ l - ~
,n
in E (G~) for n = 1,2,...
this sequences is decreasing, i.e. f
~ x~,n
hence there is in ~(G ~) f = o-lira f =
f
x~,n+l
n = 1,2,... and bounded by ~ ,
- 142 -
this p.f. f will be the image of the element x E G + by the m a p p i n g s , i.e. we put f = =~(x).
O b v i o u s l y ~ may be extended to all elements of G by letting ~ ( y )
= ~(y+) -
~(y-), ~ y e G ,
where y = y+ - y- with y+ = y v O ,
y- = y A o .
= ~ ( y + - y-)=
The so defined
mapping from G into E is one-one, order preserving and operation preserving (cf. Goffman [33 ). Hence any s~turated archimedean lattice group G can be embedded isomcrphically into the vector lattice ~ (GH) of all p.f's generated by its lattice G~ of carriers. 4.
Let B be an
abstract L-space. We may assume that the norm on B satisfies condition
(K), for on the other case the norm ##xH can be replaced by the equivalent norm = IIx+#I + Ifx-#I
IIxl/~ =
. Let B~ be the Boolean ring of all carriers of B. We fix a generalized
weak unit ~ = [ e i ,
ieI,
el>. 0 } in B, then for every x ~ a B ~ and every e i there is
a uniquely determined element x i = (el, x K) 6 B +, so called projection of x K on el, defined by xi-= (ei, x ~) : sup { Obviously we have m(e~
and for every
z e B +,
z4
el, z H 4
x~}
O@ x i ~ ei. Let now define (cf. Goffm~n [ 3] ) ) =
IX e ill
,
i~ I
x~ E B~
m(x~) =
~ II(e i, x ~ ] iaI
where the sum is + Go if
II
an uncountable number of
snmmands is > O, then m is measuro
on B ~, i.e. (B~,m) is a measure algebra in which m is not necessarily ~-finite. It is easy to see that the image space of B into E(B ~) by the mapping ~ contains all e.p.f's of the form fa=
Z
a>.q
~j ~
with
m(~)
<
+
O0
and I~ I m ( ~ )
<
+ ~,
i.e. the vector lattice E I of all so called
integrable e.p.f's. The norm completion ~
of E I with the norm U fall = Z
I~j}m(~)
i.e. the vector lattice L(B~,m) of all integrable with respect to the measure p.f.'s of ~(B ~) is the image of B by the mapping ~ .
-
5.
143
-
Let the abstract L-space B have aweak unit e; without the loss of generality we may
assume that ~ell = "J, then B ~ is a saturated Boolean algebra and the measure m a strict positive probability on B ~, i.e. (B~,m) is a probability algebra and E(B ~) the vector lattice of all real valued random variables over the probability algebra (B~,m). The image of B into E(B ~) by the mapping ~ is the vector lattice L(B~,m) of all random variables of E(B~) with a finite expectation. Let now B be separable, then the probability algebra (B~,m) is also separable,i.e, with a character CB~ is a ~
-field K of Lebesgue measurable subsets of the s e t ~
that the probability space ( ~ ,
= ~@
~), then there
R: 0 g ~ 4
1~
such
K, p) where p is the Lebesgue measure, is a set theore-
tical representation of the probability algebra (BH,m); i.e. if all X a K
% ~o
J is the ~ - i d e a l
of
with p(X) = 0, then the quotient Boolean algebra K/J (K mod J) is isometric
to B~ and if we define ~ (X/J) = p(X) for every class X/J in K/J with X ~K, then the probability algebra (K/J,~) is isometric to (BK,m) (cf. Kappos~ 5~ and ~ 6 ~ 3.4 ch.2 and 4.1 ch.3). Obviously, the representation vector space L (B~,m) of B is then isometric and lattice isomorphic to the concrete L-space L ( ~ , case the theorem of Kakutani (cf.
4
K, p). Hence we have in this
theorem 8). Generally, let B be not separable,
them B ~ is also not separable and there is an ordinal n u m b e r ~ > o ter of B ~ is ~
~ ~o
such that the charac-
• Let
o
be the cartesian product, Where every factor
A subset Z c ~ # is called
a Lebesgue cylinder if there is a representation of Z as a
cartesian product Z
where
Z~ =
=
P Z5 0 ~ ~ E~ for all ~ except a finite set ~1' ~2''''' ~n
Lebesgue measurable subset of E,
for which Z j is any
j = 1, 2,...,n. Let now S be the set of all Lebesgue
cylinders, F~ the smallest subfield of subsets of ~
containing S, then we can define
a measure (so called product measure) p on F~ as follows : p~ (Z) = m(Z~l) m(Z~2) ... m(Z~ ) n ~) Regarding the notations in this Nr. cf. Kappos [5~ and ~6] .
-
144
-
for every Z6S, pp can be extended in a well known way to F@ and to A~ smallest ~ -field A~ of subsets of n~ containing
, i.e. to the
F@ and moreover to the smallest O--
field A# of subsets o f ~ # (so called Lebesgue product ~ -field A@ ) containing A~
which
satisfies the condition : (I)
If X -c~q~and Xc-Y, where T_CA~ with % ( Y ) =0, then X m A ~
. Obviously ( ~ ,
Ap, p~)
is a probability space. The following theorem is true (Kappos [6~ ch.3
[email protected]). For the probability algebra (B~,m) with the character CB~ space (JZ~, % ,
~ ), where
K
is a ~ -subfield
=~>l~othere is a probability
of the Lebesgue ~ -field A@, satisfying
the condition (I), which represents (B~,m) set theoretically; i.e. if of all X~K/3 with
p~(X) = O, then the quotient ~-algebra ~ / J
J is the ~ -ideal
is isomorphic to BK
and if we define ~(Y/J) = p~(Y) for every Y/Je K~/J then the probability algebra (K~/J, ~) is isometric to (B~,m). Obviously, LI(B~,m) and therefore B is isometric and lattice isomorphic to the concrete L-space L( 4 '
K~, p~ ), which can be considered as a L-subspace of the concrete
ce L (~q#, Ap, p~).
L-spa-
-
145
-
References 1.
Carath@odory,
C. : Entwurf fur eine Algebraisierung des Integralbegriffs,
Sitz.#
berlcht. Bayer. Akad. der Wiss. 1938 pp. 28 - 67. 2.
Carath@odcr21C.
: Mass und Integral und ihre Algebraisierung,
Birkh~user-Verlag Basel
1956, also translation into English, Chelsea Publishing Company, New York,N.Y.1963. 3.
Goffman. C.: Remarks on lattice ordered groups and vector lattices. I. Carath@odory functions, Trans. Amer. Math. Soc. Vcl. 88 (1958) pp. 107 - 12o.
4.
Kakutanil S. : Concrete representation of abstract L-space, etc. A~n. of Math. Vol. 42 (1941) pp. 523 - 537.
5.
Kappos t D.A.: Structurtheory der Wahrscheinlichkeitsfelder
und -R~ume, Ergebnisse
der Math. und ihrer Grenzgebiete, Neue Folge, Springer-Verlag,Berlin-GSttingenHeidelberg 1960. 6.
Kappos, D.A. : Probability Algebras and Stochastic Spaces, mimeographed lectures given at the Catholic University of America, Washington D.C., 1963-64.
7.
Kappos I D.A.
: Remarks on the representation of probability fields and of spaces of
random variables. Colloquium on combinatorial methods in probability theory. August 1 - 10, 1962, in Aarhus, Denmark, pp. 84 - 89. 8.
Papan~elou. F. : Concepts of algebraic convergence and completion of abelian lattice groups and Boolean algebras. Doctoral dissertation. University of Athens, Greece (in Greek, English abstract). Bull. Soc. Math. Gr@ce. N.S.3, Fasc.2 (1962) pp. 26
9.
-
114.
Papan~elou, F. : Order convergence and topological completion of commutative lattice groups. Math. Ann. 155 (1964) pp. 81 -107.
-
Hans G. Kellerer:
146
-
Extension of stationar~ pr9cesses
Let at, 0 ~ t ~ to, be a relatively stationary stochastic process, i.e. the distributions of a t 4 , . . . , a t a n d
at4+h,...,at~+h are
identical if 0 ~ ti,ti+h ~ t o for 1 ~ i ~ n (n arbitrary). Then under the additional assumption (~) a t weakly continuous for 0 ~ t ~ to, by a simple passage to the limit a stationary extension at, t ~ O, can be constructed from solutions of the analogous discrete time problem. Giving up the condition (~) the existence of such a process can be proved using the following general theorems: 1. If ~ is the class of all finite subsets T of ~ , ~ ) with max T - min T ~ to, then to every family of consistent distributions PT over R T, T~ ~, there exists a stochastic process at, t ~ O, such that the random variables at, t ~ T, have the distribution PT ( T ~
arbitrary).
2. If at, t ~ O, is
any stochastic process, then under a
boundedness condition (which for instance is fulfilled in the case of identically distributed at) there exists a stochastic process at , t ~ O, such that the expectation of a bounded continuous function f(at; t ~ O) is always confined by the lower and upper limit of E(f(at+h;
t ~ 0)) for h ~ ~.
-
Renewal Sequences
-
and t h e i r A r i t h m e t i c
David.
1.
147
G. K e n d a l l
The K a l u z a propertv/.
In 1928 T h . K a l u z a
(8) n o t i c e d that if the formal p o w e r - s e r i e s
U(z)
=
~
(I)
u n zn
n>~0 h a s real c o e f f i c i e n t s
satisfying
Un-1
the c o n d i t i o n s
% o
u o = I, U n ~ 0 r un
(n ~
I)
(2)
Un+ 1
t h e n the formal p o w e r - s e r i e s
F(z)
=
~-
fr zr
(3)
I + u(z) P(z)
(4)
r~l d e f i n e d b y the i d e n t i t y
U(z)
:
will also have real n o n - n e g a t i v e
coefficients.
to
(2), but this is u n n e c e s s a r y ,
for if u I = 0 then (2) implies that U(z)
=
0, so that then f r
=
0
K a l u z a a d d e d the extra c o n d i t i o n u I >
f o r each r. If this d e g e n e r a t e
e v e r y u n > 0, and the c o n d i t i o n s
=
case is e x c l u d e d
(2) then m e r e l y assert that u o
=
1
0
1 and F(z) then
and that
Ul/U o, u2/u I , u3/u 2, .... is an i n c r e a s i n g
sequence
of s t r i c t l y p o s i t i v e
rate of g r o w t h of this s e q u e n c e
and we could,
The s i t u a t i o n b e c o m e s more i n t e r e s t i n g sequences
u n : n = 0,1,2,...
U n
----
un - Uo
=
. un - ~n-1
terms.
There is no limit i m p o s e d on the
for example,
if we c o n f i n e
o e o o o
u1 Uo
~
=
e
n2 .
our a t t e n t i o n to b o u n d e d K a l u z a
Because Un_ I Un_ 2
have u n
B
-
148
-
and because the factors in the last product never decrease with increasing n, it is clear that the boundedness of
u n : n = 0,1,2,...
implies that each ratio Un/Un_ 1
must be less than or equal to unity, and so a Kaluza sequence is bounded if and only if it satisfies the condition 1 = u o~
uI ~ u 2 ~u 3~..
~ 0
(5)
(We have expressed the final statement here in such a way as to include the degenerate case U(z) = 1, F(z) = 0.) Each of the power-series U(z) and F(z) will now have a radius of convergence of at last unity, and we shall have~- fr ~ 1. Thus
(Lamperti,12) ever2T
bounded Kaluza sequence is a renewal sequence. Plainly not all renewal sequences are bounded Kaluza sequences; (1,0,I,0,1,...b
the renewal sequence
illustrates this fact. On the other hand, the Kaluza theorem enables
us to write down renewal sequences which are not otherwise easily recognised as such; for example,
(~, 2-t, 3-t, @-t,...)
,
(6)
where t is any non-negatlve real number. The question then arises
: can we characterise
probabilistically those renewal sequences which are bounded Kaluza sequences? We shall give such a characterisation in this paper, by first constructing a theory of infinitely divisible renewal sequences.Encouraged by this, we shall then investigate the arithmetic of renewal sequences in general. 2. Renewal sequences and regenerative phenomena. When we speak of a renewal sequence~u n : n = 0,1,2,...Swe have of course in mind the following situation;
(~
,~
,pr) is a probability~pace
is a sequence of events(elements of the s i g m a - a l g e b r a % k pr(
~ j=l
on w h i c ~ E n : n = 0,1,2,... ) such that E o = ~ ,
and
k-q Enj )
=
pr(Enl)
-j=q ~K
pr(Enj+l _ nj)
(7)
for all k = 2,3,... and all increasing sequences nl,n2,...n k. We interpret this situation by inventing a regenerative phenomenon $
~
which is said to 'happen' at just
those times t for which the sample-pointoO lies in the measurable subset E t of the sample-space
~.
The requirement (7) can then be described informally b y saying that
the mechanism responsible for the occurrence of the phenomenon ~ suffers a complete
*) It has until now been usual to call ~ confusing to speak thus of ~
a regenerative or recurrent event. We find it
as if it were an element of ~ .
-
loss of memory whenever the phenomenon ~ The corresponding renewal sequence
14.9
-
occurs'.
u n : n = 0,1,2,... }
is then defined by
(n : 0,1,2,...)
u n = pr(E n)
(8)
Here it is clear that u o = 1 and that 0 g u n g 1, but renewal sequences also satisfy more complicated non-linear relations,
Um+ n ~ u m u n
of which
,
(9)
proved by noting that Um+ n
: pr(Em+ n) >
pr(E m ~ Em+ n)
=
pr(Em)Pr(En) ,
is the simplest (and will frequently be useful to us later). If we write fr = p r ( ~
happens when t = r, and not at any earlier t > 0),
then we shall have fr&
)
0 (r = 1,2,...
and
~- fr g 1,
(10)
and u n = fn + fn-lUl + fn-2U2 + "'" + flUn-1 Conversely, {u n : sequence; ing =~
En:
if ~ fr :
(n ~ 1)
(11)
r = 1,2,... # is given and satisfies the conditions (10), and if
n = 0,1,2,...}
is defined by (11) and b y u o = 1, then this will be a renewal
that is, we can set up a model of a regenerative phenomenon ~ n = 0,1,2,...
on a probability-space
(~
,~
by construct
, pr) in such a way that Eo=
and (7) is satisfied, and (8) holds. On introducing the power-series
(1) and (3),
one finds that the recurrent relation (11) (with u o = 1) is equivalent to the identity (#). All this is of course entirely familiar;
it is set out here merely in order to
establish the notation. It may also be helpful to the reader if we recall the very simple proof of Kaluza's result. F r o m (#), when n ~ 1, we have n o
~
un
-
~un_j
9=,
fS
-
150
-
and fn+l
=
Un+l
-
j=~l Un-j+l fj
so that n
Unfn+ 1
=
(Un+lUn_ j - UnUn_j+ 1 ) fj
~-
j=l
Now we need only consider the case in which ratio Um+l/U m is non-decreasing
uI >
Infinitel~
sequences
divisible renewal sequences.
~ un :
an>
0, and the
for increasing m. The last identity together with fl =
=u I ~ 0 then shows recurrently that each fr 2 3.
0, and then every
n = 0,1,2,... ~, a n d ~
0.
We write ~
for the class of all renewal
o for the class of infiniteIy divisible renewal
sequences defined by the requirement that (1, ~ , is to belong to ~
u2t, ...)
(12)
for each real non-negative
the first place, such sequences exist;
In
(6) in fact provides us with the example
1
(n = 0,1,2,...
n+l
Un =
t. This is a very natural definition.
).
Next, it is known (Kingman, 11) that (A)
if
u n' : n =
~ n d ~ u n'' : n = 0,1,2,...~
0,1,2, ...~
are renewal sequen-
ces~ then so is
fU u'' n (B)
:
if ~ Un(k)
n = 0,1,2,... :
~
:
n = 0,1,2,... ~
Un(k)--~ v n a_~s k--~ ~ ,
is a renewal sequence for each k, and if
for each n, then
~v n :
n = 0,1,2,... ~
is a re-
newal sequence. This being so, it will be clear that if T denotes the set of non-negative t such the sequence
(12) is a renewal sequence,
containing t = 0 and t = 1. The c l a s s ~
real numbers
then T is a closed additive semigroup
o is that subclass o f ~
defined by the require-
ment that the semigroup T is to be the half-line ~0, ~ ). From the closure of T it is clear that T = ~ 0, ~ ) of T~ In particular,
a renewal sequence
if and only if 0 is a cluster-point
~ u n : n = 0,1,2,... ~ belongs to ~ o
if and
1.51
-
only if
1/m
the sequence of mth roots
~n
:
-
n = 0,1,2,...
is a renewal sequence for
every positive inseger m. We shall co~tent ourselves here with characterising
~o'
and we defer for the time be-
ing the wider question of just what closed semi-groups T can be associated in this way with a general renewal sequence. If { u n : n = 0,1,2,...~
is a bounded Kaluza sequence, then so is (12) for each t1> 0.
Thus Lamperti's observation shows that if 9~ 1 is the class of bounded Kaluza sequences, then
5~1
'
~
~o
.
.
.
~
(13)
.
This suggests that we may best reach an understanding of the place of ~ 1
in~
by con-
sidering it as a subclass of ~ o. Our principal result is that, apart from some technical details which we shall meet presently, ~ 1
and~o
are essentially identical.
We conclude this section by recalling briefly the proofs of (A) and (B); (B) is most easily proved by use of (11), but (A) has an illuminating probabilistic proof which will provide a pattern for later calculations. Let ~ ' and
~''
be regenerative phenomena ,!
defined on two different probability-spaces, n
=
and le~
~ u n'
: n = 0,1,2,...
~ and ~ u n :
o,1,2,... ] be the associated renewal sequences. Construct the direct product of the
two probability-spaces, (~
'x~
'',~
'×
S'',
pr' x
pr''),
and on this define a new regenerative phenomenon if and only if E~
~
' and
× E~' . Then
~
~
'' both
' n
~ ' n ~ '' which 'happens' at time t
'happen' at time t; i.e. if and only if
~ '' will hav~
{ u n' u n'' • n = 0,1,2,...
( ~ ', ~
'')
~ as its renewal
sequence. 4._~. The reduction of ~ o t_~o ~ of the vector
~u n :
o'
" We find ourselves embarrassed by the zero components
n = 0,1,2,... }
, but within
~o
there is a simple means of gett-
ing rid of them. Let us put I v n : lim m~co
u nl/m
1
ifu~>
O,
=
(1~) 0
if
un
=
O.
-
Then by (B), the sequence
~v n
:
152
-
n = 0,1,2,... ~ belongs to ~
, and indeed to 5~ o.
We may have u n = 0 for each n ~ 1, in which case v n = u n for every n. If we leave this case on one side for the moment then we cam write N for the first integer n ~ 1 for which u n>
O, and then (from (9)) we shall have VkN = 1 for every k. From (7) we now
obtain ( for the ~
associated with
vn :
n = 0,1,2,...
)
pr(EkN+r) = pr(EkN n EkN+r) = pr(EkN)Pr(Er) = O, if 0 ~ r ~ N. Thus ~ u n~
recurs almost surely after N steps, and
0
wheh
N J n,
un = 0
when
N#
n,
n = 0,1,2,... } we have a zero-free member of ~ o if we put
and so in ~ u nf t
Un
(15)
~Sn "
We have used here the selection principle (Kingman, 11) (C)
if
~ u n : n: 0 , 1 , 2 , . . ~
{Ukn
:
n = 0,1,2,...3~ ~
Like (A), this property of ~ We now see that ~ o (i) (ii)
~
, then for every k S
0.
is best proved probahilistically;
i.e. via (7).
has been partitioned into
the special element (I ,0,0,0,...); equivalence classes of renewal sequences,
each class being characterised
by a unique zero-free infinitely divisible renewal sequence ~ u nf n = 0,1,2,...~ We w r i t e r
defined as at (15).
~ for the set of equivalence classes at (ii), identifying each member o f ~
with the associated zero-free infinitely divisible renewal sequence ~ u ~
: n = 0,1,2, ....
This allows us to work with zero-free infinitely divisible renewal sequences only, from now on. (To obtain the general member of ~ o
other than (1,0,0,0,...) we have only to
choose a positive integer N and insert (N - 1)-ads of zeros into ageneral member o f ~ . ) If ~ u ~
:
n= 0 , 1 , 2 , . . ~
~
then it is natural to write
-x n un = e
(n = 0,1,2,...),
(16)
-
where x o = O, and 0 6 x n ~
~
lows that the system ~ o f
of C
all vectors ~ = { x n : n = 0,1,2,... } is a proper pointed con-
phenomenon.
corresponding to u n = I (all n ), i.e. to the almost
We can interpret
as lying in the vector space R ~
then implies that C
-
for all n. From (A) and the infinite divisibility it fol-
vex cone, the vertex (0,O,0,...) sure regenerative
153
with
(B) in the present situation if we think
~he topology of simple convergence,
for (B)
is closed.
As usual we obtain a base for the cone by slicing it by a closed hyperplane which avoids the vertex;
the hyperplane
defined by x I = I is convenient.
convex, closed, and metrisable
The base ~
so obtained is
(for RC°is such), and we shall now see that it is com-
pact. For (9) tells us that (on ~) Xm+n <~ Xm
(17)
+ xn
and so on the base ~ we must have
O<xn
<'=1
=
n.
(18)
Thus
.z~ [0,~
,
n~O and the compactness
now follows from the theo2em of Tychonov (the cartesian pro-
of
duct of compacts is c o m p a c t ) ~ 2 ~ s i g n Accordingly
for cartesian product~
(~; for a simple proof see ! ) ~
which is non-empty and is a ~ j bitrary element x of this cone ~
possesses a set ~
of extreme points
, and if y denote a %-Fpical extreme point then an arhas a Choquet representation @f the form
£ xn
=
4.
where/~ x is a totally-finite
yn
~x(d~)
(n = 0,1,2, ..... )
measure on the Borel subsets of
We shall identify the elements of 8 ~
(19)
~ J~ .
and so make (19) explicit,
thus obtaining the
explicit (and, as it turns out, faithful) parametrisation
un the measure ~ sequence
=
exp
-{~2#5y n ~u
(~)~
(n = 0 , I , 2 , . . .
being u n i q u e l y determined by the z e r o - f r e e
{ u n : n = 0,1,2,
... ~
in
~ 'o"
)
i~finitely
(20) divisible
renewal
-
~,
The relation between
~I
and ~
154
-
o"
We have already remarked that (1,0,0,...) is the only bounded Kaluza sequence having zero elements. L e t ~ Theorem
~I denote ~
I with this exceptional element removed. We now prove
I.
The class ~ I
of zero-free bounded Kaluza sequences is identical with the class
o
of zero-free infinitely divisible renewal sequences. We have only to prove that ~ o '
Proof.
c-- ~ 1
Suppose then that { u n ! n = 0 , I , 2 , . . . }
belongs to d~o and define x as at (16) above. Then for each non-negative real number t there will exist non-negatlve real numbers fr(t) unity,
such that
oo ~-
(r = 1,2,...) with sum not exceeding
£ e
zn =
[ 1 -
n=o
fr(t
)
Zr ~ - I
(Jz/ < I )
(21)
r=l also e - t x n - ~ I as t -~ 0
Obviously fr(0) = ~ r l ;
for each n, and so, because the quan~i-tx n ties fr(t) are polynomials in the quanti~±es Un(t) = e , it follows that lim ~0
fr(t)
=
fr(0)
= ~rl
"
We mew assert that
1-fI ( t ) lim
-
t-~O
xI
(22)
t
and %hat fr(t) lim---~ t~0
-
2Xr_ 1
-
xr
-
Xr_ 2
( r = 2,3, ....
)
(23)
For a proof of (22), and of (23) when r = 2, we have only to recall that
-~I fl(t)
=
e
-tx2 and that
f2(t)
=
e
-2txl -
e
.
The resul~ (23) for general r is then easily established b y an inductive argument based on the identity n-1
f n = (1 - f l ) + f 1 ( 1
- um_ 1) -
which is simply a rearrangement of ( 1 1 ) .
fj Un_ S
-
(1-u m)
(n~ 3),
-
155
-
N o w fr(t) > 0, and so (23) tells us that ~ is a concave vector ,i.e. that
Un
un-2
2
~
~-I
( n = 2,3,...)
ao that the last condition in (21 is satisfied and ~ u n z n = 0,1,2,... ~
belongs to
t
1' as required.
(That the quantities u n are positive and bounded and that u o = I is
of course obvious.)
6. The extreme r%Ts of the c o n e
~
•
We now change our point of view, and study the analytical object tain a better understanding of the probabilistic object base~
of the cone ~
~o"
~1
in order to ob-
We have seen that the
is just the set of vectors x specified b y
x o = o, x I = I, x n ~
0,
~2x n ~
0 ,
(2~)
and we shall show (as is presumably well known) that the extreme points of the convex set defined (2@) are precisely the vectors y(k) where
Yn(k)
n
i
=
( 0 ~ n ~ k ) (25)
k
( k ~ n )
and k = 1,2,...,oo. We note that (241 implies that Xm+ n 4 x m
+
xn, that X n ~
n, and
that x n is non-decreasing for increasing n. That the y(k) are extreme is easily shown; suppose t h a t ~
is some extreme point of
different from all of these. Then for some (finite) positive integer k we shall have Yn
=
n for n = 0,1,2,...,k
and k L Y k + I ~ k+l
(26)
N o w put
=
y
+
~-I
(y
_
~(k)),
where 0 < p < I and q = 1-p. Then ~ = px + q~(k), and we shall have contradicted the extreme character of ~ u n l e s s
x fails to satisfy (24).
(Note that x and ~(k) differ at
least in their (k + 1)st components.) But ~ ~ ~(k), because these two vectors agree up to n = k, and y is non-decreasimg.
Thus x must fall wo be conoave, an~ oan only do so
on the triplet (k-l, k,k+l). We must therefore have
2 ( Yk - %Tk(k)) < (Yk-1 so that
2yk
< Yk-1
+
Yk+l
+
q~k-1 (k) q
whenever
+
(Yk+l - qYk+l (k))
'
0 x q ~1. N o w let q--b0, and recall that
-
156
-
itself is concave; we find that we must have 2y k =
Yk-1
+
Yk+l' so that Yk+l = k+1,
so that Yk+l = k + I, which contradicts (26). Thus no such ~ can exist. The integral representation (19) now takes on the simple form
S
14_ j , ~ m
"
where the coefficients ~
~(J)
+
oo~(ee)'
(27)
are non-negative real numbers such that ~ - ~ j
<
eo. The ~ 's
depend on x, and are uniquely dete~mimed by x, as the following evalutions show:
~S
=
A2 xS_
-
1
(S
1,2,...
=
)
(2s) (29)
We can thus re-write (27) in the completely explicit form
=
xl , d = ) *
.4 2 xj_1 r Z ( = )
- ~(J))
.
(30)
lz_J<~ O n assembling the preceding results we obtain Theorem II. A n infinitely divisible renewal sequence
ium
: n = 0,1,2,... ~ is either the degenera-
sequence ~1,0,0,O,... #, or it admits a unique representation of the following form: for some positive integer h, and for some real non-negatlve un
=
0
~ 's satisfying ~- ~ j L oo,
when h does not divide n and
n/h j=l
S
> n/h
w h e n h does divide n. If we put Vn(S)
=
exp(-~n(j)) , for j = 1,2,... co,
then when h = 1 we can write this relation more concisely in the form u =
I I j=l
v(J)
J
,
the powers and product being formed componentwise. h >I
(32) The more general case in which
can then be obtained from this b y interpolating (h-1)-ads of zeros.
When h = 1, there are elegant formulae corresponding to (28) and (29) which express
-
157
-
the ~ 's terms of the u's. The first of these, /~j =
log
Uj-lUj+l 2
(J = 1,2,...
)
(33)
u3
brings out very clearly the connexion between the Kaluza property and the fact that the 's are non-negative. Similarly the formula for
~
11u~ v7 j ~.1
:
log
=
lim j*e~
}
UJ-lUj*l
log (u~/uj+ 1)
(D'
,
(3@)
shows that the non-negative charcter of ~ ~
reflects the non-increasing character of
an infinitely divisible renewal sequence. This last formula also shows that if ~ = then uj/uj+q --~ I, while if Z e o ~
0 then uj/uj+1--~ e~c° > I, and then Un--~O geometri-
cally fast. Thus ~ is transient if ways persistent when ~ o o then we find that U n ~
0
~
>
O. It is however not the case that ~
= O. Thus, if we put ~j = S / j 2 for j = 1,2,...
C(~) / n ~ , and s o ~
is al-
and 2
= 0
is persistent in this case if and only if
~1. As for nullity, it is easily verified that Un-~O if and only if one at least of the following conditions holds:
~ ~j = ~,
~qm > o.
(35)
~. Infinitel2 divisible regenerative phenomena. Np to this point we have attached the adjective 'infinitely divisible' only to the renewul sequence itself, mud not to the regenerative phenomenon ~ described by it. But if we identify regenerative phenomena on different probability spaces provided that they have the same renewal sequence, then we can say that ~ is infinitely divisible preciselywhem
in this sense it is equivalent 6o n
....
n
for each positive integer m. Here, for each m, dent copies of a regenerative phenomenon
mm
'
~ m k (k = 1,2,...m)
(36)
are to be indepen-
~ m' and the regenerative phenomenon (36) is
to be understood to 'happen' when and only when every one of ~m1' ~ m2''''' 'happens'.
~mm
-
158
-
The variety of infinite divisibility envisaged here is quite different from that occurring in the classical theory of Khintchine, L@vy, and Kolmogorov, and it is also quite different from the infinite divisibility of point processes studied by Goldman (5), Lee (13), and Matthes (15). In fact it is clear (c.f. Grenander (7)) that i f ~ e i s used to denote any associative product for random variables of a~y kind, then we can say that such a random variable is infinitely divisible if it is distributionally equivalent to the ~ -product of m independent copies of some other random variable, for each m =2,3,.. We obtain the classical theory when the random variables are real-valued, and ~ denotes addition. We obtain the kind of infinite divisibility usually considered for point processes when ~ denotes the formation of the set-theoretic union of the ~wo realised sets of points° Ne obtain a dual notion of infinite divisibility for point processes, not so far as I am aware studied by previous writers, when ~ d e n o t e s
the formation of the
intersection of the two realised sets of points, and it is this interpretation (in the special context of regenerative phenomena viewed as 'indicators' of point processes) that we are concerned with here. 8.
The regenerative phenomena associated with the extreme rays of ~ .
The canonical decomposition (32) for zero-free infinitely divisible renewal sequences suggests that we should examine the character of the regenerative phenomena associated w~th the irreducible factors v(1), v(2),...v(~ ). We lose nothing of interest by restricting attemtlon to the zero-free case, for the degenerate renewal sequence (1,0,0,...) corresponds to a ~ which almost surely never recurs, while an infinitely divisible renewal sequence for which h > I corresponds to a ~ whose recurrence times are almost surely multiples of h. The simplest case is that of the infinitely divisible renewal sequence v(oo). Here and later it will make the notation more easily comprehensible if we introduce an appropriate positive finite % (in this case
~),
and then put e
= ~.
Thus we shall in the
present case discuss the ~ for which the renewal sequence is
(Vn( ~ ))-logp
(n = o,I ,2,...).
But this is just (I,~,~2,...), where o ~ ~ I ,
and so ~
is fully described by saying
that it recurs immediately after an occurrence with probability ~, and that when it once
-
stops
159
-
'happening' it stops forever. In the notation of
F(z) = ~ z ,
~ 1, U(z) = (1 - ~ z ) -1
and
so that the (defective) recurrence-time distribution just has mass p at t=1.
The next simplest case is that associated with v(1). Here the renewal sequence to be considered is (1,2,~,~,...), where as before 0 ~ F(z) = ~ z / ( 1 - ( 1 - ~ ) z ) ,
so that ~
z 1. Here U(z) = 1 + ~ z/(1-z) and
is now persistent,
and has the recurrence-time
distribution
pr(~ : r) : ~ (I - ~)r-1
(r = 1,2,...).
Finally we must consider the more complicated case when the basic renewal sequence is v(J) with I L
j 4 ~.
un=9
n
This time we have to describe the ~
(o~ n~
j),
un
F S (~
n)
for which
(o L ~ ~I).
We shall then have
u(z) =
1-~Jz j
+
1-~z
pJz j 1 - z
and now
,(,) :
+
{
I+
o2z ÷
m
(37) In this case also,
therefore
~ is persistent, but the recurrence-time T for ~
now
has a rather complicated distribution which we can describe as follows.
(i) (ii)
With probability jo , T = 1.
(s = 0,1,2,...), variables
TI'
~
m ~-- ~ i ' where pr(m=s I T > 1) = ~ J ( 1 )s i=1 and where (given that T ~ I and that m = s) the random
If T ~ 1, then T ~ 1+ j +
~2'''''
qZm are conditionally independent and distribu-
ted over the possibilities 1,2,...,j with probabilities proportional to
1, ~ , . . . ,
p j-1
To make a model for the regenerative phenomenon with a general zero-free infinitely divisible renewal sequence as at (32) we have only to const~met the direct product of a countable sequence of probability-space carrying regenerative phenomena of the types just described (with appropriate
~ 's in each caseJ,and then to identify ~
simultaneous occurrence of each the component phenomena.
w i t h the
-
9"
160
-
The Central Limit Theorem for remo_~al sequences.
We are now going to construct an analogue of the Central limit Theorem (in its general (triangular array) form (6)) for renewal sequences. First we need some notation. We shall write u for a renewal sequencer m~ : n = 0,1,2,... ~, ~
for the corresponding
sequence of t th powers, and lim ~(r) for the termwise limit of a sequence of renewal sequences ~(r) (r=-1,2,...) as r ~ o o ce). We shall write u ~
(by (B) of §3, this will again be a remmwal sequen-
for the renewal sequence whose n th terms is UmVn, and similar-
ly for any larger number of factors. We note that anF sequence of renewal sequences ~(r) contaimma convergent sub-sequence; this is proved by an obvious compactness argument depending on the fact that 0 mean an arty
Un(r) ~1 for all r and n. By a triangular array we shall
~(1,1 ), ~(2,1), ~(2,2), u(3,1), .Oo
of
~(3,2),
~(3,3),
0 0 0
renewal sequences, there being i entries in the i th row (i = 1,2,...). We shall this
a null array when
lim
u1(i,j) = I,
(38)
$~e limit being umiform with respect to j(1 ~ j W i). By the i th marginal product of a triangular array we shall mean the renewal sequence z(i)
=
~(i,1)~(i,2)~
... ~u(i,i),
(39)
and we shall say that a triangular array is convergent (necessarily to a renewal sequence) when lim ~(i) exists. We are now going to prove the basic Theorem III. If a null triangular array of renewal sequences is convergent then the limit is infinitely divisible, and either the limit is (1,0,0,0,...) or each of the terms is positive. Also, every positive infinitely divisible renewal sequence can be obtained in this way. Proofs Let us prove the second (easier) half of the theorem first. If ~ is a positive infinitely divisible renewal sequence, then we can exhibit it as the limit of a null triangular array by putting
- 161 -
Un(i,j)
=
VnYi
(j
=
1,2,... i~
similarly if v is (1,0,0,0,...)
i ~ 1);
then we can put n
Un(i,j)
=
exp (- ~ - F ) ,
and we again obtain a null triangular array with v as limit. We now turn to the first half of the theorem~
and suppose that we are presented with a
convergent null triangular array, with limit v. The first step is to prove that either (i) the limit is (1,0,0,0,...)
or (ii) every V n > 0. It will suffice to deduce from
v1= 0 that Vm+ 1 = 0 for all m ~ l ,
because we know that if v 1> 0 then Vm+ I ~ ~1 +1 >
We make use here of an inequality~
Ur+ s ~
1 + uru s
-
max
-
um
0.
noted by Kingman (11):
(4o)
(Ur,Us).
From this it follows that urn+1 .< 1 + UmU 1
= 1 -
(1 - U l )
~
1 -
(1 - U l ) U
~
.
Now
if 1 - & ~ • .4 1 , provided that the positive number 6 is small e n o u ~ ,
and so since we
shall have
1 -~
< u1(i,j ) .4 1
for all j g i, provided that i>~ I ( ~ ) ,
we cam conclude that
o ,~ u m + l ( i , j ) ~ ~ / U l ( i , j ) " for all j ~ i provided that i @ Im, for an
(41) fixed m. But this implies that 0 ~ Vm+l{ V~l,
and so the desired conclusion follows. From this point onwards we can without loss of generality assume that each v n is positive,
and it only remains for us to show that v
is infinitely divisible.
~)This is most easily proved by noticing that Ur(1-u s) 4~ 1 - Ur+ s (an easy consequence of (7)) and then appealing to symmetry.
-
162
-
In the first place it is clear that if n is fixed then Um(i,j) > 0 for m = 1,2,...,n and 1 ~< j ~ i, provided that i exceeds some i o (which we henceforth suppose to be the case). Accordingly (using (11)) we can write Vn(i)Vn_2(i)/Vn_1(i) 2 in the form i ' ~i
1 + A(i'j) + fn(i'j)/un-l(i'j)
j=l where
]
,
(42)
1 + B(i,S) n-1
A(i,S) = h = ~
and
fn(i,S)Un_h(i,J)/Un_l(i,j)
-
(1 - fl(i,j))
,
n-1 B(i,~) = Z fh(i,j)Un_h_l(i,~)/Un_2(i,~) h=2
if n ~ 3. From (38) and the inequality 1~
-
m
Um~
uI
we know that Um(i,j)
uniformly for 1 g m ~ n and 1 ~ j ~ i, while f r o m ~ 1 - fl(i,S) each tend to zero as i - ~ ,
and
(I - fl(i,~))
---~ 1 as i ~ ®
,
fr g 1 and fl = Ul we know that
fr(i,j)
uniformly for 2 ~
r g n and 1 g j g i. This being so, the
numerator and denominator of the jth factor in the product (42) will be of the form I + z, with I z| g 1/2, for all sufficiently large i, and so can be written in the form exp(z + ~ z 2) where]~l g 1, It follows that
log
I Vn(i)Vn-2(i) ~ v~_1~ J
/~
j~__ill fn(i' S) ' Un_ 1(i,S)
(A(i,j) + fn(i,j)/Un_l(i,j))2
+
- B(i,j)
(A(i,j)- B(i,j))
>
Y j=l
fn(i,J)
M(I - u1(i,S)) ~
Un_l(i,J) (43)
for all sufficienly large i, where M depends on n only. Now O ~
u1~
1 and so u lg
e-(1-Ul ) . As
i
V-T S=1
u1(i,j ) = v1(i )
-- v I> o ,
-
163
-
it follows that i
(I - u~(i,j)) j=l is bounded as i-~ co, and therefore (using (38) again) that i as i ~ ~ . 2_ (1 - ul(i,j)) 2 - - ~ 0 j=l Accordingly
the inequality (43) implies that VnVn_ 2 v2 n-1
log
is non-negative;
2 i.e. that VnVn_ 2 - Vn_ 1 ~ 0 for n = 3,4,5, . . . .
Now this inequality
must in any case hold when n = 2, in view of the identity u2 = f2 + ~ which holds for all renewal sequences. Thus
~v n t n = 0,1,2,...I
~ ~
= u2 '
is a positive bounded
Kaluza sequence, and so is infinitely divisible by Theorem I.
10. The arithmetic of renewal sequences~ ) The convolution semi-group of probobility-laws tion seml-group of their characteristic
on R 1 (or, equivalently,
functions)
has exceedingly interesting arithme-
tic properties which have been unravelled by L@vy, ~ n t c h i n e , series of investigations.
(See, for example,
the multiplica-
and other in a classical
chapter 5 of Linnik's book (14).) We shall
now make some general remarks which suggest that there may exist a similar of renewal sequences',
'arithmetic
and then we shall proceed to construct it.
We shall conTine ourselves in this discussion to aperiodic renewal sequences; we shall suppose
that the set of values of n for which U n ~
0 has unity as its grea-
test common divisor. This will have the effect, in particular, sequence (1,0,0,0,...)
from consideration
that is,
of eliminating the
(it obviously has every other renewal sequen-
ce as a factor) and of restricting the infinitely divisible renewal sequences to the class of those which are zero-free. We lose very little by this limination,
because
~) In my Loutraki Lectures section 10 of this paper was replaced by a sketch of a similar
'arithmetic'
presented in (10).
for general
'delphic'
semi-groups.
An account of this will be
-
164
-
i f ~ u n , n = 0,I,2,3,... } is periodic with period h = g.c.d.
I n , Un>
0~>
2, and if
u = v,w is a factorisation into two renewal-sequence factors, then we can confine our attention to those values of n which are divisible by h, and then in virtue of property (C) of ~ 4 we shall obtain a corresponding factorisation of the aperiodic renewal sequence ~nh
~ Unh : n = o,1,2,... } into the renewal sequences
~ Vnh : n = 0,1,2,... }
and
: n = o,1,2,... } , both now necessarily also aperiodic.
It is k n o w n (Chung, 3) that every renewal sequence u can be identified with the sequence of 'top-left-hand-corner'
elements
~ p(~): n = 0,1,2,... } for a Markov chain in which
we can take the state I to be aperiodic when the renewal sequence is such, and then (Kendall, 9) we know that we can write
un = ~
where ~
e ine
~(de)
(n = 0,1,2,...),
(44)
is a probability measure on the Borel subsets of the circumference C of the
unit circle which is symmetric about the diameter from
@ = 0 to e =or , and which
(apart from a possible atom at e = o) is absolutely continuous with respect to Lebesgue measure. It follows that we can
identify the double-sequence
( •.. ,u 2 ,u I ,uo,u I ,u2, •.. ) w i t h the characteristic
'function' of the probability law H in such a w a y
that the
-multiplication we have been considering for renewal sequences corresponds to the operm~ion of convDlution for the associated measures ~
~ in other words, renewal se-
quemees with the operation of ~ -multiplication form a system which is isomorphic with a sub-semi-group of the convolution semi-group of probability laws on C. (This assertion is true even if we relax the requirement of aperiodicity; is to permit further atoms to ~
the only effect of periodicity
, located at the h th roots of unity°)
This fact makes it worth while for us to look for a factorisation theory of renewal sequences, but it does not guarantee the existence of one, and while it may suggest true theorems it does not enable us to dispense with the necessity for proving them. This is because we do not know how to c h a r a c t e r i s e t h e respond b2 wa~ of formula ( ~ )
sub-semi-group of measures H which cor-
to renewal sequences~u n : n = 0,I,2,... ~
ry conditions on ~ have already been mentioned,
. Some necessa-
and while they can be added to (Poller
-
165
(~) has shown that in the aperiodic case ~ h a s
-
a density which is continous away from
e = 0),even these augmented conditions are , as we shall see, not sufficient. For let 0 < r < 1,and consider the continous positive probability density ~'(e)
=
(2~)-1
1 - r2 1+2r cose + r 2
(0 g e ~ i~r),
which has the require~ symmetry. We can re-write this as
(2~)-1
{ I + Z n-1
(-r) n
(e i n e -
e-ine)}
and so
£
f
vn = Je C
in@
~ (de)
= J
e/he ~'(@) d@
=
(-r) ~
(n = O , q , 2 , . . . ) .
C
Obviously this is not a renewal sequence, but ~ = v ~ v is a renewal sequence (indeed, is an infinitely divisible one). Thus the f a c t o r i s a t i o n ~
=
~ ~ ~
(where /~ corres-
ponds to u) is meaningful for the factorisation theory of probability laws but is irrelevant to the theory we wish to construct. Accordingly, while we shall Imitate the classical proofs wherever possible, we shall frequently have to support them by arguments specially designed for the present situation. W e start b y showing that there exist indecomposable aperiodic renewal sequences;
that
is , aperiodic renewal sequences u with the following two properties:(i) (ii)
u { (1,1,1,...); if
~ = X ~,
then E = (1,1,1,...) a n d w
= ~, or vice versa.
W i t h the notation of (1) and (3), let fl > O,
f2 = 1 - fl > O,
fr = 0
(r = 2,3,...),
so that Ul = fl Then
and
is persistent and aperiodic,
u2 = f~ + f2 " and its recurrence-tlme distribution is limited
to the two values T = 1 and T = 2, each of which carries positive probability. Suppose if possible that there is a proper decomposition ~ = ~ w ,
so that each of v and w
is an aperiodic renewal sequence distinct from (1,1,1,...)
. Let
~ ' and
~ '' denote
-
the two associated regenerative phenomena. tify
~
with
~'
n
~
} amd
-
Then, as previously explained, we cam iden-
''. This shows that each of
so that their recurrence-time r = 1,2,...
166
distributions
~
' and
~''
are not defective;
let us write
~ h r : r = 1,2,... } for these. Now fl = g l ~ '
and h 1 is positive;
must be persistent, [ gr
:
so that each of gl
also each of gr and h r must vanish for r > 2 beeausenwe
c~nnot have
T ~ 2. Finally g2 and h 2 must be positive, because g2 = 0 implies that gl = 1 amd so that v n = 1 for all n, and this is excluded. From fl = Ul = VlWl = glhl amd 2 2 + g2h2 + g2h~ + g ~ 2 fl2 + f2 : u2 = v2w2 : ~1hI we obtain glhl + g2h2 + glh2 + g2 h2 = fl + f2 = 1 However we also know that
(gl ~ g2)(hl + h2) = 1, and so glh2(1 - gl ) + g2hl(1 - h 1) = 0, and yet this expression is equal to glg2h2 + hlh2g 2 > 0. Thus the c o e f f i c i e n t s i n
the expansion of
(1 - flz - f2z2) -I form an indecomposable
(45)
aperiodic renewal sequence when fl + f2 = 1 and fl and f2 are
positive. This example is by no means exceptional.
In fact the next theorem will show (for ex-
ample) that if { u n : n = 0,1,2,...] is an aperiodic renewal sequence such that u n un+ 1 for at least one n, then either ~ is itself indeccmposable,
or it has an inde-
ccmposable proper factor.
Theorem IV. If the aperiodic renewal sequence then either it is indecomposable
~ u n : n = o,1,2,...~
is not infimitely divisible,
or it has an indecomposable
proper factor.
-
Proof:
Let u be am aperiodic
renewal
167
sequence which is
se further that in every proper decomposition proof that u is infinitely . I ~
If in a renewal
-
divisible.
not indecomposable,
and suppo-
neither factor is indecomposable.
We must
We first prove the elementary
sequence u n = 1 for some positive
n, say for the first time
when n = m, then u n = 1 whenever n divides n, and otherwise u n = 0. This follows from Kingman's
inequality
(40), which with r + s = m shows that either
u r = u s = 0, or u r = u s = I. The second possibility or vice versa,
can only occur if r = 0 and s = m,
and so:aeither m = 1, or m > 1 and u n = 0 for 1 ~
that the recurrence-time
n~
m. It is then clear
T is almost surely equal to m, and the truth of the lemm~ is
evident. On applying the lemma in the present than unity for every positive I,...)
is trivially
context we see that we can assume that u n is less
n, for we have supposed u to be aperiodic,
infinitely
divisible.
and u = (1,1,
We can recall that b y hypothesis ~ is aperio-
dic and so is not (1,0,0,0,...). N o w let m ~ l
be such that 0 4 u m 4 1, and let m be fixed until further notice.
the functional
X(u) b y um
so that
is positive
=
that X ( B ~ )
and finite
~
,
of ~ implies
(46)
.
= X(~) + X(E).
on proper components
that v m > 0, and as they must be aperiodic the possibility
,
exp ( - X ( u ) }
0 < X(u)~
and we observe
(47)
It is
relevant
here that the functional
like
~, and are not allowed to be (1,1,...),
v m = 1 is excluded b y the lemma. Accordingly
every proper decomposition
a division of X(~) into a finite number of finite positive
small.
the functional
Then we can construct
converges
We
~' for which
Consider for a fixed proper component ~ of ~ the infimum
of X(E) , where w ranges through all components infimum is positive.
parts.
functional.
first assert that each proper component ~ of ~ has a proper subcomponent is arbitrarily
X
E of ~, for obviously they must be such
We shall employ X(~) in place of what Linnik (1~4) calls Khintchine's
X(~')
We define
to its infimum,
of E. Suppose
a sequence
if possible
that this
of proper components
and from this sequence,
of ~ on which
using compactness
and
-
168
-
(B) of ~3, we can pick out a sub-sequence which converges to a component E' of ~ such that X(~') is equal to the infimum. But E' is clearly a proper component of ~, and so of ~, s@ admits a proper decomposition,
and on exploiting this fact we can contradict
the infimum character of X(~'). We now consider all possible decompositions
: y(1)~
~(2)~
...
~z(k)
(48)
into exactly k proper components. We may as well assume that the components at (#8) have been arranged in an order of increasing X-values, and in relation to the set of all such decompositions we write ~
(obviously positive and finite) for the supremum of
X(~(1)), Compactness ersures that this bound is attained, and we suppose that i~ is attained at (48). We assert that X(v(j))
= %
for all j = 1,2,...k. For if this were
not so we should have
d.,= x(E(1)) for some i such that 1 %
.....
x(~(i)) 4 x(z(i+1) ~ ... ~X(z(k))
i ~ k. From ~(i+1) we can peel off i components for each of
which X is arbitrarily small (but positive), to E(1),E(2),...,~(i) above ~
and if we transfer each of these in turn
then we can retain the original ordering but increase X(~(1))
, which gives a contradiction.
Thus a k-fold decomposition can be fmund in
which , for each of the k components, X(~) = X(~)/k. We next observe that the preceding arguments cam be imitated exactly when m ' ~ m'' ~ 1 , and both 0 ~ um, ~ 1 and 0 ~ Um,, ~ 1, if we use instead of the functional X(~) the modified functional
Y(~)
log
I urn, ~m''
If X' and X'' are the X-functionals based on the choices m = m', and m'',then obviously Y = X' + X'', and so we are able to conclude that if um, and Um, , both lie in the open interval (0,1) then for each k it is possible to find a proper decomposition o f ~ k factors ~(k,j)
whenever j ~ k.
(j = 1,2,...,k)
in such a way that
into
-
169
-
We now consider the triangular array
~(2,1),
~(2,2),
~(3,1),
~(3,2),
ooa
~(3,3),
oe.
@@o9
eoo
,
and note that the marginal products ~(k,1) ~ u(k,2) ~
o..
~u(k,k)
are all equal to B, so that the array converges to ~. If then we can show that the array is a null array, it will follow from theorem III that the l~m~t ~ must be infinitely divisible, and the proof of theorem TV will be complete. That the array is a null array is a consequence of the fact that ~ h a s
been assumed to
be aperiodic. We recall that for any renewal sequence the set~n : U n >
0 ) is am addition
semi-group, and from the assumed aperiodicity of u we know that in the present instance this semi-group has unity as its greatest common divisor. There will therefore exist in the semi-group a finite set sl,s2,...,s t o~ positive integers without a common prime factor, and as all sufficiently large positive integers can be written in the form
als I + a2s 2 + ... + ats t, where the a's are non-negative integers, it follows that all sufficently large positive integers belong to the semi-group. In particular, therefore, we can find positive integers m'' = m and m' = m+l such that Um, and urn,, are both positive, and as before we can assume that each of these quantities is less than unity because otherwise the infinite divisibility of u would be immediate. We may therefore suppose the triangular array to have been constructed with such a choice of m' and m' ', and accordingly it is a feature of the array that
um,(k,S) -~ 1
and u m,,(k,S) ---I
as k--~oo, uniformely for j = 1,2,...,k, where the essential new feature of the situation is that m' = m'' + I. We now appeal once again to Kingmsm's inequality (~D), using it to obtain
o ~ ur(1 - u I ) 4
I -Ur+ ~
-
170
-
as a universal inequality for renewal sequences. From this we deduce that
I
-
u1(k,j) g
~-Um,(k,j) um,,(k,J)
,
(49)
and hence that u~(k,j) tends to unity j-uniformly as k - ~ o o ,
i.e. that we have a null
array. Theorem IV has therefore been established. We now prove a ~haerem w h i c h asserts that every aperiodic renewal sequence can be built up out of infinitely divisible and indecomposable factors in at least one way (we make no assertion about mnicity).
Theorem V. If.~ is any aperiodic renewal sequence, then we can always write
= Z(I) ~Z( 2)~...~E
,
(50)
where each E is indecomposable, ~ is infinitely
divisible, there are not more than
countably many E's, and the ~'s or w may be absent. Remark:
It will appear from the proof of theorem V that a factorisation (50) always
exists in which w has no indecomposable factor. Proof:
If ~ is infinitely divisible we put S = ~ and omit the ~'s. If B is indecompo-
sable we put E(1) = B and omit E and the rest of the E's. We can therefore exclude these two cases. Then, by theorem IV, ~ has an indecomposable proper factor. The compl~mentary factor may be infinitely divisible, we arrived at the desired decomposition.
or indecomposable,
in either of which cases
If this is not so, then the complementary fac-
tor has an indecomposable proper sub-factor. Rather than say 'and so on', we now use Zorn's lemma. We consider the system of objects S specified as follows: -
(i)
each S is a mapping from the set of all indecomposable aperiodic renewal sequences v to the set (0,1,2,...);
(ii)
for any positive integer k, for any choice of the k distinct indecomposable aperiodic renewal sequences ~(I),~(2),...,~(k), the renewal sequence
and for any S,
-
s(z(1))
171
-
s(~(2))
v(1)
S(z(k))
~(2)
~
...
aperiodic
renewal
.v(k)
is a factor of ~. N o w for any indecomposable n ~ 1 ; this implies
~) that S(~) has a finite
fixed v. We partly-order if S'(~) ~ S''(~)
(attained)
for each indecomposable
aperiodic
1 for all
least upper bound for each
the system of objects S by asserting
of objects S has the Zorn property: system.
sequence v, we must have V n ~
that S' ~
S'' if and only
~, and we then observe that system
each chain of objects S has an upper bound in the
This being so, there exists a maximal S; let us choose one,and fix it through-
out the followin~
argument.
Because we may without
loss of generality
suppose that u is not infinitely
we can take it that 0 d u m ~ I for some m ~ 1; and with this the X-functional
as before.
X(~) must be finite and positive
divisible,
value of m we construct for each ~ for which
S(~) ~ O, and so there can be only countable many such v's. Let us label them as ~(1),
v(2), . . . . Each product ~ s(y(~)) Z(1) ~
s(~(2)) ~(2)
is a factor of ~, and may be identical construct
~ in this way,
mentary proper factor, so (by compactness) r-~oo.
...
~Z(k)
with ~. We have
nothing
where ~(k)
~(k),
if k - * o o
Z(1)~v(1)
occurs S(~(k))
if it were indecomposable extended,
say. We cannot be sure that limw(k) in some suitable
~
sequence
(1,1,1,...),
sequences.
associated
Because
but it must do
(kl,k2,...) , where k r ~
~
as
sequence w such that
times.
Because ~ is a factor of ~ it must be aperiodic,
or possessed
an indecomposable
and
proper factor then S could be
Thus, by theorem IV, w is infinitely
divisible,
and
of ~ has been found. Of course it may t-am out that ~ is
in which case it could be omitted.
We have now completed dic renewal
exists,
... ~ v ( 1 ) ~ v ( 2 ) ~ . . . ~ w ,
and would not be maximal.
the desired decomposition
left to prove if we can
and so we may as well suppose that there is always a comple-
In this way we can be sure that there exists a renewal
u=
res p
S(z(k)) *
the construction Our results
of the promised factorisation
imply that the sub-semi-group
as at (~4) w i t h aperiodic
as usual we understand
renewal
aperiodicity
sequences
to exclude
theory for aperio-
of probability
measu-
u must have a convolu-
the possibility
u=(1,0,O,..).
-
172
-
-tion-arithmetic which is very similar to the convolution-arithmetic
of all probability
measures om C. This suggest that in seeking to charaoterise such special probability measures
~
one might usefully look for properties of the full convolution semi-group
which are not shared by the sub-semi-group. As all the positive characteristics we have looked at are shared by both, it may be more profitable to look at negative ones, and to ask how far the pathologies of the classical convolution-arithmetic
occur in the pre-
sent contexts we shall not, however, discuss such questions further here.
11. Postscript:
The delay to pedestrians croosin~ a road.
Consider a pedestrian wishing to cross a road carrying a single lane of traffic. We suppose that it takes a(finite) time c to cross the road when it is free of traffic, and that the vehicles (whose size will be neglected,
and whose speed is conventionally
taken to be unity) form a Poisson stream with i n t e n s i t y %
. If we are given that it is
possible to cross at time zero, then the chance that it will be possible to cross at a later time p(t)
t is readily seen to be =
exp ~ - % m i n
(c,t)#
(0 ~ t Z co);
(51)
here 0 < c < oo. The possibility of crossing is evidently a regenerative phenomenon in continuous time, and the function (51) is the continuous-time analogue of the renewal sequence. Regenerative phenomena
~
in continuous time, and their associated p-functions,
have been intensively investigated by Kingman (see 1_~I for a s ~ , ~ a r y o f
his work). The
factorisation problems treated in the present paper are being separately studied in the p-function context, and an account of this work will be published elsewhere (Kendall, 10). The role of the particular p-function (51), for 0 & c & co, corresponds exactly to that of the renewal sequence mi~(~,n)
Un =
9
(where j = 1,2,...,oo),
(n = 0 , I , 2 , . . . ) in that it is infinitely
(52) divisible and corresponds to an ex-
treme ray of the associated Choquet cone, so that the most general infinitely divisible p-function can be canonically expressed as an integral-product involving the functions (51). Obviously we can provide a similar traffic-theoretic
interpretation for (52) when
j is finite, if we allow pedestriam and vehicles to move discontinuously in discrete
-
173
-
time, and this interpretation may be found of some interest in view of the fact that the recurrence-time analysis of §8 proved to be rather complicated. It may be remarked parenthetically that the recurrence-time analysis associated with (51) is equally complicated. For a study of this see Tanner (17). We can interpret the infinite divisibility of (51) or of (52) either by dissecting (i.e. sorting out the cars according to, say, colour, or first name of driver), or by considering the problem of crossing several equally wide lanes of traffic without waiting between them, the time- and space-zero for each lane being adjusted appropriately for the effect of the crossing-time c. (When pedestrians are allowed to cross one lane and then wait on an 'island' until they can cross the next, the problem is more cpmpllcated (Mayne, 16_).) The multi-lane traffic problem is of especial interest because it supplies a concrete realisation of the composition of point-processes by intersection rather than by union.
-
174
-
References 1.
F.F. Bonsall, Soc.
2.
'On the representation of points of a convex set', J. London Math.
38 (1963), 332-334
G. Choquet,
'Les c ~ e s
(Stockholm,1962),
convexes faiblement complets',
Proc. Interm. ComEr. Math.
317-330.
3.
K.L. Chung, Markov Chains with Stationar 2 Transition Probabilities (Berlin, 1960).
4.
W. Feller,
'On the Fourier representation for Markov chains and the strong ratio
theorem' 5.
J, Math, Mech. 15 (1966)
273-283.
J.R. Goldman, Stochastic Poimt Processes : Limit Theorems and Infinite Divisi~llit~ (Thesis, Princeton University, 1965).
6.
B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables
(English translation, Cambridge, 1954).
7.
U. Grenander, Probabilities on Algebraic Structures (New York, 1963).
8.
Th. Kaluza,
't~ber die Koefficienten reziproker Potenzreihen', Math. Zeit. 28 (1928)~
1 61-170.
9. D.G. Kendall,
'Unitary dilations of Markov transition operators', in Surveys in Pro~
bability and Statistics (ed. U. Grenander) (Stockholm, 1959). IO.D.G. Kendall,
'Delphic semi-groups, infinitely divisible regenerative phenomena, and
the arithmetic of p-functions',in preparation. 11.J.F.C. Kingman,
'An approach to the study of Markov processes', J. Royal Statist.
Soc. (B) (to appear). 12. J.Lamperti, ly
65
'On the coefficients of reciprocal power-series', American Math. Month(1958), 90-94.
13.P.M. Lee, Infinitel~ Divisible Stochastic Processes 1966)o
(Thesis I Cambridse University,
-
175
-
14. Yu.V. 14~I~, Decomposition of Probabilit~ Distributions (English translation, Edlmburgh, 1964).
15.
K. ~at~hes, 'Umbeschr~xkt teilbare Verteilumgsgesetze statiom~rer zuf~lliger ~uktfolgen',
Wiss. Ze!t. Hochschule fGrElqo~rotechnik Ilmenau 9 (1963),
235-238. 16.A.J. ~ayme, 'Some further results in the theory of pedestrians and road traffic', Biometrika 41
(1954), 375-389.
17.J.C. Tammer, 'The delay to pedestrians crossing a road', Biometrika 38 (1951), 383-392.
-
176
-
Optimal Bounded Control with Linear Stochastic Equations and Quadratic Cost
Eustratios Kouaias ~
(Talk delivered at the international conference, Probabilistic methods in Analysis held at Loutraki Greece between May 22 and 4, 1966)
Abstract A method is presented for the computation of optimal control for linear stochastic discrete systems when the control variable is a bounded scalar. The main part of the paper deals with the contimuons time problem. The cases with and without penalty on the conrail variable are studied separately, and expressioms for the optimal policy, and the average cost obtained. The best linear policy is investigated, the steady state solution found and the average cost for the best linear policy calculated. Finally the average costs for the best linear policy and the optimal policy are compared. 1. Introduction There is a considerable body of literature about the optimal control of systems with linear, stochastic or non-stochastic performance equations. The special problem of this paper is mentioned by Bellman 1961 and 0rford 1963. For simplicity we have assumed the state, the control and the random input are of one dimension and that the state variable can be measured precisely and without delay. 2. Discrete Time Case Suppose we have a control object (system) the behavior of which in discrete time is expressed by a first order difference equation, i.e., xt+ I = p.x t - u t + et
(1)
where x t is the state variable at time t, which is assumed to be observed precisely and
~) This is a s~mmary of a paper which will appear in the Journal of Mathematical Analysis and Applications.
-
177
-
without delay, u t is the control input at time t, which is applied immediately after the state variable x t has been observed, e t is the random input at time t, due to some uncontrollable factors. The random inputs are assumed to be independent identically distributed random variables (I.I.D.) with mean~and
variance ~ 2
, both finite,
p is a known non-negative constant i.e., p ~ O. To be more realistic we suppose that the control is uniformly bounded i.e.,
lutl~
a .
t = 0,1,2,...
(2)
where a is a positive constant. We suppose that the process begins at time t = 0 and at that time we only know s
X 0
Definition
=
X
•
: Every sequence of numbers (Uo, u1,..~Un_l) ~ p u n] such that
t = 0,I,...,n-1,
] ut~ ~ a,
is called admissible control sequence. Let U n denote the set of all
aam4ssible control sequences. If we have n steps to make and the controller wlshes the state of the system to be as close to zero as possible i.e., x t = 0 if possible for t = 0,1,2,...,n-1
then a r e a -
sonable cost function is s
Jn(Xo, [un])
=
~
(x~ + ~
(3)
e o,...,en_ 1 L t=O
where E denotes expectation over the join@ distribution of eo,el,...,en_l,
[Un] denotes
the sequence of control values which are used from the time we begin the process i.e., t = 0 until the end of the process, a n d S ,
is non-negative constant.
Our task is to find the optimal cost
Vn(X o) = ~ n
JnCXo, [ Un])
(~)
- 178 -
for all admissible control sequences. Applying Bellman's principle of optimality and taking into account our original model we obtain: 2
lUol %a lUol ~
which minimise (@) is called optimal
control policy; Vn(X o) is called optimal cost; and the values of the state variable Xo, x~,...,X n attained during the optimal control is called optimal trajectory.
Computational procedure Lemm~ I i) ll) lil) iv)
The opt~m~l policy ~o(X) is - ~ f o r m l y
continuous and increasing w.r.to x,
p.x - Uo(X) is uniformly continuous and strictly increasing w.r.t, x, dVz(x) aT
~ V~(x) exists, is uniformly continuous and strictly increasing w.r.t, x,
the optimal policy rule and the loss function can be found by iteration using the relations : V~(x)
=
~
(6)
and either :
2~
~o
= E ~V'n_l(PX - Uo + e)]
if l~o(X)l ~
a
(7)
or ~o(X) = ~a
if (7)
results in
Uo(X) ~ a
or
Uo(X ) ~ -a
then V~(x) = 2x + pE[Vn'_l(PX - U o
v)
+ e)] ,
Vn(X ) is strictly convex w.r.t, x, for each n = o,1,2,... , and can be calculated by Vn(X ) = K2 + ~ 2
+ E[Vn_l(p x _ ~o + e)]
Vo(X) = x 2
.
-
~. Continuous
179
-
Time.
If instead of t~klng the time ~n~t equal to one as in (~), we take it to be at, then the corresponding
equation will be xt = P xt-at - mut +
where again x t is the state variable u t is the control
independently
after x t_ mt has been observed.
identically
so that ~t is a Wiener process. p = I -
where T is a positive
~at,
constant,
In a sense, u t is associated reasons
at time t,
input applied immedla~ely
a ~t are the random inputs, v a r i a n c e ~ 2 mr,
~t
lutl
We also make the reasonable
assumption
au t = u t ~t
i.e.,
T > 0.
with the speed we apply the control,
of being more realistic,
Then X t = (I - o a t )
distributed with mean p m t and
and we require,
for
that:
~a,
xt_~t - u t ~ t
+
A~t
"
Thus in the limit as A t - ~ 0 we obtain
d--~ + Tx(t) The process ~ equation
'
~
+ u(t) =
(8)
' must be understood
(8) has a well recognized
in a generalized
sense (Yaglom,
1962), but
formal interpretation.
~.1. The Cost Function Definitionz stitutes u(s), u(s)
Any piecewise
on admissible
continuous
control.
function u(s)
such that
Denote by U(t) the class of all admissible
0 ~ s ~ t. lf the cost incurred
at time ds is (x2(s) + ~ u 2 ( s ) )
are the values of the state and the control variable
determine
lu(s)l ~ a, 0 4 s 4 t concontrols
ds where x(s)
and
at time s, then our aim is to
the optimal cost t
u~U(t)
~
(x2(s) + ~u2(s))
ds
,
(9)
o
where ~ a positive
constant,
value of the state variable
and the corresponding
at time s. Referring
optimal policy G(z,s) where
to linear stochastic
equation
z is the (8), Bellman's
-
180
-
(Io)
principle of optimality leads to
" ~
= x 2 - (~x-
p)~+~-
6-2
~u2(x,o)
- u(x,o)BV~x~ t) )
I u(x,o)l .~a V(x,o) = 0. A g a i n if we were working with a given policy u(z,s) rather than the optimal ~(x,o),(10) would be valid hut without the "min" operation. First the optimal policy rule and the differential equations for V(x,t) are formulated and then the cases of no control a = 0, unbounded control, a = ~ ,
and bounded control
0 < a < oo, are studied the last case being the most interesting. Finally the best linear policy is derived and compared with the optimal policy rule. ~.2 The ootimal policy rule and formulation of the equations for V(x~t~. F r o m (10), the optimal policy rule will be: ~(x,o) = a
£(x,o) G(x,o)
= ~
if
1
= -a
~V(x,t) Bx
r~v(x.t)) "
~x
if
BV(xot) ~x
~
if
2 a
}~V(x't)} < ~x
.K
2 aX
(11)
-2 a
Since the continuous time case was approached as the limit of the discrete time case, we postulate here that V(x,t) is convex w.r.t,
x, a fact which was stated in lemmR (1)
and is proved rigorously by Kounias, 196#, therefore DV(xot) ~z
is increasing w.r.t.x.
Thus there are functions wl(t) , w2(t) such that
8V(x,t) ~x
] ~~ xt
~
2 a~
if
x>
w2(t )
~
2 a~
if
w1(t)_~
x _~ w2(t)
(12)
181
-
-
The optimal policy rule ~(x,o) becomes ~(x,o)
=
a
~(x,o)
~
g(x,o)
= -a
if
x ~ w2(t)
BV(x.t)
if
if
wl(t) @ x @ w2(t)
(13)
~w1(t),
x
~(x,o) ~ _
By ~(x,o) we mean the optimal rule at the present time when the present value of the state variable is x. Of course this optimal rule depends on the time remaining until the end of the process. Now putting (13)into (10) we obtain the equations for V(x,t) i.e.,
~v + ~ x2 - (~ x -/~)~-
~v(x.t)~t :
~~av +
~(~(x,o))2
~v - ~(x,o) Ei
(1~)
w i t h bounds._.-~ and matching condition a) V ( x , o )
=
0
b) V(x,t) continuous on the lines x = Wl(t) and x = w2(t) c) ~V~(~'t)continuous and takes the value 2 a ~ f o r
x = w2(t) and - 2 a ~
for
x = wl(t) 3.~
The case
a
=
0,
~
>
0.
The only admissible policy is u(z,s) = 0 and the solution of ( 1 4 ) will be
vote,t)
:
1 - 2~ e
+
( t-
2
2(1
+
-
-
)2 x +
e -~t,) 1 ,- e - 2 ~ t ~ + " 2~
(t
1 - e -2~t -
),
2~
0 _~t -oo ~X400
We o b s e r v e
here
that
for
t large
) +
(15)
-
x2
e~2
v~
Vo(X,t) = - - + - @ ~ +
(--+
2~-
~
182
2
)t-
2~
-
1_
(¢2+
2~
) +o(1)
(16)
2r
and
lira ~VO ( z , t ) t~oo ~t
6~2 -2~ -
=
H2 + %2
(17)
Obviously for a=O, wl(t) and w2(t ) coincide and we find the function w(t) such that
a Vo(x,~) 0
for
x = w(t)
•
~x Thus
w(t) = : e T
( 1 - e-~¢ 1 +e -~)
and w(t) can be regarded as the limit of wl(t) , w2(t) as a ~ O 3.4
The case
a
=
~
I
A>O
In this case (10) can be minimized without restriction,
5Vco (x,t)
_ x 2 _ 1__ ( BVco (x.t)) 2 - (~x - H )
#~
~t
Voo(X,o) =
~x
therefore
B Ve ° + ~2
~ 2Vo °
t~O
ax
~ -~ x
-oo <x
2
(18)
O,
with solution V~(x,t)
= gl(t) x 2 + 2 ~ g 2 ( t ) x
+ g3(t),
(19)
where g1(t),g2(t),g3(t) , can be easily found by equating the coefficients powers of x in (18) and then solving ordinary differential corresponding
loss function for A = 0 is identically
of the
equations. Notice that the
zero. The optimal policy rule is
~(x,o) = 1_ (2 g l ( t ) x + g2(t)) 2p 2~
(20)
For t large we obtain
-r~ + ~T2%2 +:X 3[ +
7 2 Z 2 + :~
A
+/,
(21)
- 183-
and
~(x,o)
= -~
+
V~2A2
+ R x +~ -rA +
7r ~.e.,
approximately
lim t-~o 3.5
linear w.r.t,
BVoo(X,t)
~r2A2 + 2A2 + k
+ o(1)
x. Also, the average cost is defined as 2
= (-~
+
~t
(22)
V~
~
2
. ~) 8 2 + ~
~2A2 + A
.
(23)
The case 0 < a 0
Since V(x,t) is known for a=0 and a=oo, we have found a lower and upper bound for the loss V(x,t),
v ~ (x,t) ~ V(x,t) ~ Vo(x,t) . Voo(X,t) , Vo(X,~),
(24)
are given by (19) and (15), therefore, V(x,t) is O(x 2) for l a r g e ~ ,
uniformly in t, and is 0(t) for large t, uniformly in x. 3.6
Steady State Solution
We postulate that lim 5V(x.t) t~
= c
independent of x, i.e., we postulate
function V(x,t) can be represented approximately
that for large t the loss
as
V(x,t) ~ ct + h(x). Also, it is assumed that lim w1(t) = w I and l i m w 2 ( t ) = w 2 t~oo t~o ' where w 1 and w 2 are independent of t. Thus for large t the optimal policy becomes
G(x,o) = a ~(x,o) = !_ 2/
~V(x.t) ~x
u(x,o) = -a The corresponding
if
xhw 2
if
Wl~ x ! w2
if
X!Wl
(25)
.
equation for V(x,t) is
c=x2 _ ( ~ _~) ~v(x.t) ~x
.2__ 2
~2v(x.t) %2
Z~2(x,o) +
-
G(x,o)
~V(xot) Bx
(26)
Equation (26) for x _>w2 or x <_ w 1 can easily be solved w.r.t.
V(x.t)
as an ordinary linear differential equation. The constant of integration is 3x determined from the fact that V(x,t) is 0(x 2) for large x. In the region w I <_ x _< w2,
-
184
-
(26) is a Riccati equation and to get an approximation use the results of Kalaba 1959 who gave the solution as the limit of a sequence of functions which convergenses quadratically.
So we have four im~nowns c, wl, w2, d where d is the constant of inte-
gration for the Riccati equation. To determine wl, w2, c and d we have four equations i.e.,
BV(x°t) =
2~a
at
x = w2
=
- 2 ~ a at
x = wI
(27)
Dx V(x,t) ~x
If the bound, a, is sufficiently large, then from (27) we get /2
~
if2
+o(
e_B(r~ + ~
1
k)2 a 2 )
a3
Wl= (~,~+ ~ / ~ )
a-
A
/J
~/~2-A2 +Z ~ - 2 +~
+
o(-~-a )
+
0('1a3)
(28)
~ 2 ~ 2 + }~ where B = So the average cost c decreases exponentially with a 2 as a ~
for ~ ~ O, which is not
the case f o r ~ =0 4. The Best Linear Polic~ If instead of the optimal policy we try to find the optimal linear policy i.e., the best on policy of the form U(x,o) = y
(29)
V(x,t) as defined by (9) for all the ad-
missible linear bounded policies, we proceed as follows: observe that since the policy u=O belongs to the class of admissible linegr policies and for u=O, Vo(x,t) is given b y (15), then the loss function V(x,t) which corresponds to the best linear policy behaves at most quadratically w.r.t, x, and linearly w.r.t.t for large t.
We again
study the steady state solution for which ~(x,o) = a ~(x,o) = ~ x ~(x,o) = -a
+ 6
if
yx +SA
if
lyx +61
if
~x
a ~ a
+ 6 ! -a
where ~ and 8 constants to be determined. We postulate also that then the steady state equation resulting from (10) is
lim t-.co
d__V(x,t) dt
= ~;
- 185 -
-- ~
- (~ x - H ) ~
+-2 87
+ ~
2(x,o) - ~(x,o) %~ a-~
(30)
~x + 6
rule ~(x,o)
with the conditions a) ~V(x,t)~x
is continuous on the lines
=
a where the control
changes form b)
V(x,t) is O(x 2) for large Ix I.
The resulting constants of the integration for the form of the solution in the three regions x~
a6 - - ,
-a
- 6
a
-
6
--a
~x~
and
- S
x
and the average cost ~ are determined by the above conditions completely. Thus ~ is givem as a f~sation of ~ a n d ~(~,6)
w.r.t., ~ a n d
6,
6 which are still ,,n~.elm. If we minimize this function the values of
and ~ corresponding to the best linear
policy are obtained. Thus _ T
=
=
Yo ~o
+
~o
2
g2
a2
~o
1
o(~e
>
I
+
+
o(~e
- ( ~ + /o) ~o 62
22 ) a2 -
2 = G2 ( -
q-h+
}/E2~2 + ~ ) -
+A
c~A
1 + 0 (a~- e
where - ~
+ ~2Z2
+
2
H(-~-Z
)
J +V~2~ 2 ÷ Z )
=
~o
(31)
(32)
V/~2 Z2
+
Z
.
C onclusiom By comparing (28)
and (31) we see that the average costs c and ~ for optimal Ann op-
timal linear rule are of the same order, i.e.,
-
2
o(a-~-
e
186
-
y~ )
so that for all but small a ~he best linea~ policy (29) has a performance scsu~el~ distlnguishable from that of the unrestricted optimal policy.
Acknowledgement: This research was supported in part by the Air Force Office of Scientific Researnh, Contract No. AFOSR
@9(638) - 1302.
-
On
187
-
a Fourier Transform in Infinitel2 Many Dimensions Povl Kristensen,
Lars Mejlbo
and Ebbe Thue Poulsen
I. Introduction Schwartz' on
Rm
space
~n
of infinitely often differentiable rapidly decreasing functions
can be characterized in an abstract, essentially algebraic way, which cam be
formulated in cases, where
R n is replaced by an infinite-dimensional
with the resulting characteristics are called of type ~
,
space.
Spaces
and it is shown that such
spaces exist. For functions points
~a~ n
x ~ Rn
it is possible to calculate their function values
as well as the Fourier transform ~-~
(.)
an element
~
in a space of type
defined on the infinite-dimensional
product of two elements,
and
~/
at
from the abstract framework,
and it can be shown that the same holds for spaces of t y p e ~ In particular,
~(x)
.
can be represented as a function
space R e ~ .
Furthermore, the scalar can be interpreted as
in a space of type
the integral
of their product with respect to the non-existing "Lebesgue measure" in Re ~ . integral is to be understood as follows: The Gaussfumction exp ( - I[ ~ll 2)
This
is trazs-
ferred from the integramd to the measure, the remaining part of the integrand turns out to be continuously extendable to respect to the Gaussiam measure on Analogously,
R e ~ ~,
and this extension is integrated with
R e ~ ~.
the functional representative of the Fourier tr~nform 5c~
is given by ~-~(~0)
=
exp ( - i < ~ , ~ > )
where the integral is interpreted in a similar way.
~ (~)
d W
,
of an element
-
188
-
2. Notation By a scalar product on a complex vector space we understamd a Hermiteam form which is linear in the second variable, conjugate linear in the first. product in
C n is denoted by
will be denoted by
< .,. >
(.,.), or
the derived norm by
~.,.~ ,
I .I;
<.,. > ?
The scalar
other scalar products
and the derived norms by
I! .II
resp.
Ill "llJ . A scalar product on a locally convex space is called continuous if the derived norm is continuous. If S is a locally convex space with a comtimuous scalar product write the value of a linear functional
f ~ S ~
amd we defime the scalar multiplicatiom im tf,x>
S ~
at the vector
then we
< .,.) , x ~ S
f,x ~ p
as
by
= ~ (f,x
in order that the canonical embedding of
S
into
S ~ induced by the scalar product
be linear. If S 1 and S 2 are locally convex spaces, we demote by
L(S1,S 2)
the space of all con-
tinuous linear mappings of
L(S1,S 2)
will always be pro-
S1
into
S 2.
The space
vided with the topology of uniform convergence on bounded sets. The dual of an operator
T ~ L(SI,S 2)
is the operator
T ~ from S~2 into S~
defined by 4 T~f2,xl > If
S1
amd
S2
=
(f2,TXl >
for
f2 ~
$2'
Xl ~ S1"
are provided with scalar products, them two operators
T ~ L(SI,S 2) I
amd
T ~
L(S2,SI)
are called ad~oimt with respect to the scalar products if
Z.. T*:X.2,xl > For two operators
R,T
for
= 4.::Z....2,TxI >
on a linear space [R,T] =
S
x ~ ~ S~,
i = 1,2.
we define their commutator
~ R,T]
by
RT - T R .
~. The spaces ~ m . Denote Schwartz'
space of rapidly decreasing infinitely differemtiable complex valued
-
functions on
Rn
by Y n ,
i.e. jpn
189
-
consists of those functions
~(xl,x2,...,Xn)
for
which all functions
• .-xn are bounded, or equivalently,
(-~-~-~))
those ~ ,
... ( ~ x n
for which all these functions are square in-
tegrable (that these two characterizations are equivalent follows from Cauchy-Schwarz' inequality on the one hand and an inequality due to Sobolev on the other hand). Consider the operator definition of Jp n PJ
pj
qj,
and
j = 1,2,...,n,
entering so fundamentally into the
:
=
-iz~j
'
qj = xj .
For the purpose of p r e p a r a t i o m f o r -valued linear functions P and
Q
l~ter generalization we introduce the operatordefined by
P(v) = }-S vjpS '
Q(v) = ZS
vsqs
for v = (Vl,V2,.°.,v n) ~ C n. Thus , for v g R n we have
Q(v) W (x)
= (x~v) ~ (x).
Clearly, the operators P(v) and
Q(v)
are continuous linear operators in J# n satis-
f~mg [P(v), P(w)J
=
EQ(v), Q(w~]
~P(~), Q(w~
=
-i (v,w) I,
=
o,
relations which have been studied extensively because of their importance in mathematical physics. Let us now make the following formal definition: A locally convex space jo? is called a space of t y p e ~ n if and only if (i) (ii)
~?
is provided with a continuous
There exist linear mappings
P, Q
scalar product
< .,.~ ;
from C n into L ( ~ ?, ~?),
such that
-
P(v)
and Q(v)
190
are self-adjoint for v a Rn and such that
(iii)
KP(v), P(w~
=
K Q ( ~ , Q(w)1
(iv)
LP(~), Q(w~
=
-i (v,w)
(v)
there exists an element fIN/oll =
(vi)
-
I and
~o
P(v)~Yo
=
o,
and
for v,w ~
c~z
~jo?, called the cyclic element such that
=
iQ(v)~/o
for all v ~ Cn, and
RV/o is dense in JP?, where R denotes the subalgebra of L ( ~ ?, ~ ? ) generated by all operators P(v) and
Q(w), v,w ~ C n.
It is clear that this terminology is justified insofar as ~ n with the cyclic element ~/o(X)
=
is a space of type J~ n
Vro defined by exp (_I~ I xJ 2 ).
That the condition (vi) is fulfilled in J° n follows from the observation that functions of the form (x) =
exp (-1/2 I xl 2 ) °
polynomial (x)
are dense i n ~ n . Furthermore,~ n is actually a maximal space of type ~ n If ~ ?
in the following sense:
is a space of type J° n then there exists a continuous mapping J
such that J ~ o (?) =V/o (n),
of JP? into ion
J is am isometry with respect to the scalar products in Jp?
and J°n, and JP(?)(v) = P(n)(v)J,
JQ(?)(v) = Q(n)(v)J.
In particular, J is one-to-one,and J ( ~ ? )
is dense i n ~ n .
In the proof of this result essential use is made of the fact that Jp n that the topology o f ~
n is the weakest topology with which the usual L2-scalarpro -
duct and all operators @. Spaces of t~pe ~
is complete and
P(v)
and
Q(v)
are continuous.
.
Modelled over the definition of section 3, we introduce the following definition:
- 191 A locally convex space
(i) (ii)
such that
P(~)
~?
~ ?
is called a space of type
(iii) (iv)
(v)
is provided with a continuous scalar product
Q(~)
are self-adjoint for
~P(~),P(w)S
= ~ Q(~),Q(~)S
~P(~).Q(~
=
I
~
Re~,
= O,
and
-i<~,~
for
there exists an element
such that
~o'
~
~o
~.,.~ ;
from ~
into
L(~,~
?)
and such that
~, ~ ~ J ;
called the vacuum element, in
~?
wnd for all
?)
if and only if
there exist continuous linear mappings P,Q and
(vi)
G-
?
is dense in
generated by all operators
,
where
~ 7R
,
and
denotes the subalgebra of L ( g ?
and Q(~), ~ , ~ .
P(~)
It can then be shown that spaces of type
~
do exist, and a concrete, so-called Fock-
- representation of them can be constructed directly from these requirements.
However, the Fock-representation of an element
~
of a space of type
~
is analogous
to the sequence of coefficients in the Hermite-series development of a function ~ a ~ , but the coefficients are even more complicated insofar as the n'th "coefficient" function belonging ~o ~ n . an element
~
is a
Thus it is very hard to see any resemblance between such
and a single function on some such nice space as R n.
~. Functional representation In order to construct a realization of spaces of type
~
as spaces of functions,let
us first discuss how it is possible to recover the point values of elements ~
~b °n
from the abstract characterization. Consider the ~ -distribution and the translation operator The(X)
=
~(x-h)
for
q-h
defined by
x ~ R n.
In terms of these, we can write
It is easily seen that the element
8
~
n~
is characterized in terms of the type
-
192
-
_ jpn _ structure by Q(v)S= 0
for all v
~ On .
Here, and in the sequel, Q(v) and P(v) denote the continuous extensions to ~ n ' o f the same operators in ~ n
or, equivalently, the duals of Q(~) and P(~) in L ( J P n , ~ n) re-
spectively. In fact, these equations determine an element
~ E J#n~uniquely except for a mamerical
factor. Similarly, the operator T h E L ( ~ n,jpn) is characterized uniquely except for a numerical factor by the equations ~hP(V)
=
P(V)~h
~hQ(v)
=
(Q(v)
for all v e C n.
- ~ h,v>)~ h
Unfortunately, the factor involved here is a function of h. However, the requirement that T h be an isometrywith respect to the scalar product reduces the ambiguity to a factor of modulus I, and the requirement that the mapping h - * ~ h be Frechet-differentiable from R n into L ( ~ n , J pn) with the derivative characterized by
I~
T h+te~
finally determines ~ h u n i q u e l y
t=0
=
-iPCe)Th
except for a constant scalar factor.
Thus except for a normalization factor of modulus I, the function representation ~ ( . ) of ~ n
is uniquely determined by the type-~n-structure.
In the case of spaces of t y p e ~ w e
proceed analogously.
First,we note that the dual of a space of type ~
has a Fock representation, and that
the operators P(~) and Q(~) have continuous extensions to this dual, and then we prove that the equations Q(~)~
:
o
for all
~
jo
have a one-dimensional space of solutions in the maximal dual space - the dual of the minimal space.
-
195
-
These solutions can be exhibited explicitly in the Fock representation, and it can be shown that they do not belong to the dual of all spaces of type
~
.
Turning next to the analog of the translation operators, we consider an element ~ e R e ~ and seek an o p e r a t 0 r ~ ? satisfying the equations
~ Such an operator ~
QCw) =
(QCw) - < ~ , ~ / > ) T~
for all
~e~.
does not exist in every space of type6- , for instance not in the
minimal space, but when it exists, it is unique and can be normalized so as to have all the characteristics of the translation operators in ~ n . Fortunately, there exist
spaces of type ~
, in which the translation operators are
defined, and which have 6 in their dual space;one of these we have labelled ~ We are now in a position to define the functional representative
e ~
(.)
.
of an element
as the function defined on Re 50 by
The differentiability property
of "C,p implies that ~ (.)
is (infinitely often) diffe-
rentiable with directional derivative given by
t=O In fact, the operators P and Q are completely analogous to the finite-dimensional case, as is seen from the relation
But the analogy is
more
striking than this. Explicit calculation shows that the func-
tional representative of the vacuum element is
" ~ o ( ,,p )
=
exp ( -1/2 f/O0 H 2),
and this Gaussian factor is present in the functional representative of every element e ~
in the sense that if we write
~(~o)
= :FC~)
exp C-1/211~pII2),
-
then the function F on Re ~
194
-
has a continuous extension to Re ~ *
Now consider the weak distribution (or cylindrical m e a s u r e ) ~
.
~2
defined on cylindri-
cal sets in Re J~ ~ as follows: For every finite orthonormal set M = I~I' ~ 2 ' ' ' ' ' ~ k ~cRaf we consider the projection PU: R e ~
P~
=
--+
R k defined by
( ~ f , W1 > , < f , ~ 2 > , ' ' ' , < f , ~ >
),
and for every Borel set E C R k we define ~2
(p~1 (E))
= ~
exp (- Ixl2/ 2 c 2) d ( 2 ~ 2 )
-I/2 x).
This weak distribution is tight in R e ~ ~ , and hence it can be extended to a Baire
mea-
sure in ReJP * . It can be proved that if
¢(~)
=
F(~) e~(-~/2 II ~ //
V/(?)
=
G(~)) exp(-1/2 II ~f~ 2 )
are the functional representations of two elements ~ ,
~,
VI~
e
a )
~
, then
:
or, more suggestively, but symbolically
=
f ~
G(~)
exp(
-
I] ~l12) d('V-1/2
)
:
Naturally, the fact that ~ 2
is a proper measure in Red'facilitates
the proof of this
result since it makes application of standard convergence theorems possible. Let us finally turn to the Fourier transform. I n ~ n it can be characterized by the relations
for all v
P(V)J ~
=
- ~-Q(v)
Q(v)~
=
f P(v)
~ C n, and it is very simple to show that the ~nalogcus relations determine
a Fourier transform in
~
uniquely except for a scalar factor.
-
195 -
One would expect this Fourier transform to be given in the functional representation by the symboli¢ formula ||
(~)(~/)
=
M
Jexp
(-i< ~ ,VI> )~(~) d(ct~)
for some suitable real number o~ , or, more exactly, by the meaningful formula (~-~)(~)
=
f exp (-i<50
(~(~)
=
F(~)
,V/> ) F ( ~ ) p 1
(d~),
where exp(-1/2l/~0//
2)
,
and this is in fact true. The proof of this result is somewhat involved
- as can be
expected, an essential part consists in identifying the Fock representative of the linear functional E E ~
defined by
M
A change of scale shows that
>> ,
where t h e " d o u b l i n g mapping" D i s c h a r a c t e r i z e d i n t h e f u n c t i o n a l
exp (I/2 lJ~I/ 2)(D~)(~)
=
r e p r e s e n t a t i o n by
exp ( It ~ll 2)~(21/2~).
From the representation of Q(~ ) and P(U/) in the functional representation as multiplication and differentiation operators, it follows that (DQ(~) ~)(~0)
i.e. and , analogously,
D Q(~/)
=
exp (I/2 Jl~oll2)(Q(~)~)(2!/2 ~o )
=
exp (1/211~o//2) <~, 21/2~o>~ (21/2~o)
=
2 I/2 (Q('~/)D (~)(~),
=
2 I~ Q ( ~ ) D
-
D P(~)
=
196
-
2 -I/2 (p(~/) + i Q ( ~ ) )
D.
Now these equations can be studied in the Fock represent&~ion, they characterize a transformation D ~ L( 6
,~
and it turns out that
) im~quely except for normalization.
With the aid of this "doubling mapping" D, finally, the result is established.
-
197
-
Some Problems Arisin6 from Spectral Analysis by R. M. Loymes
(Cambridge)
1. Introduction A two-sided
sequence of complex-valued
random variables
(~;
n a + ve or -ve integer l
is second-order stationary if the "covar~ance function" E~Xn~mS exists and is a function of n - m only (these values are actually the covariances this is a natural
only if E ~
= 0, but although
condition to impose we shall not find it necessary to do so). For
such a sequence there exists a spectral representation Xn = ~ [ 0 , ~
in the form
einedZ(e)
(1)
where equality is to be interpreted as follows: for each n the difference between the two sides is a random variable D n w i t h EIDnl 2 = 0, which is of course true if and only if D n = 0 with probality one. The stochastic process ~Z(e):8~[0, has orthogonal increments in the sense that if the intervals
(al' 82)'
denoted symbolicallyj
(e3' e~)
E[dZ(~) ~
ted as a limit in mean-square
E[((Z(52)
do not overlap, = 0 if
~ ~,
2~) 1 appearing in (1)
- Z(e I) ) (Z(e 4) - Z(e3))
= 0
a property which is more easily
and the integral is to be interpre-
of Riemann sums.
There are various ways of proving the existence of the spectral representation
(1) ,
but one of the most convenient is to use the theory of unitary operators in Hilbert space. By generalising
the concept of Hilbert space to allow the inner product to take
values which are no longer scalars but elements of a more general topological vector space, it is possible to define sequences of "random variables", suiSable abstract space, which are second-order ral sense;
with values in a
stationary in a generalised but natu-
with certain fuzther assumptions a spectral representation
which is identical in form with (1), is possible.
of the sequence,
A sketch of this theory, which first
appeared in Loynes (1) and (2), is given in section 2 and 3, and certain problems which arise in or are suggested by its development It is not yet
are pointed out.
clear how useful this generalisation will prove to be, although it
-
clearly helps to demonstrate -order stationary processes.
the exact status
-
even of the classical
Certain applications
some of which are to appear in Loymes tain problems
198
are, however,
theory of second-
possible,
and these,
(3), are sketched in section 4; here again cer-
arise, which are pointed out in the hope that this will hasten their
solution. We shall not consider the case of a stochastic
process whose parameter
whole real line, rather than just the integers, 2.
but analogous
space is the
results continue
to hold.
LVH - spaces
Our intention defined,
is to consider spaces on which a vector-valued inner product
and it is clear that if such an inner product
is to have useful properties
the space in which it takes its values must have a suitable siderations dorff) (1)
in mind,
topological
therefore,
Z has an involution:
i.e. a mapping
= z, (az I + bz2)~ = ~ z ~ (2)
+ Sz#
ordering in Z b y writing
the topology
in Z is compatible
mentioned
conditions:
if x ~ 0 then x = x~;
(5)
Z is complete
If the following
(z~) ~
for all complex a,b;
zI ~
then we define a
z 2 if z I - z 2 ~ P;
with the partial
ordering,
of the origin,
then whenever a neighbourhood
in the sense that there such that if x ~ N e
of the origin is
it will tacitly be supposed to belong to this set { N ~ ;
(4)
(6)
the following
z~z* of Z to itself with the properties
is a basic set ~ N e ~ of convex neighbourhoods and 0 ~ y ~ x then y ~ Ne;
With such con-
if it is a complex (Haus-
there is a closed convex cone P in Z such that P~ - P = 0: partial
(3)
structure.
we call a space Z admissible
vector space (TVS) satisfying
can be
(as a locally convex TVS).
condition (6) is also satisfied,
if x I ~ x 2 ~ ... ~ X n ~
It will be observed the usual topology
then we call Z strongly admissible
:
... ~ 0, then the sequence x n is convergent.
that these conditions
are satisfied b y the complex numbers with
if we define the involution
and P to be the non-negative
operation
to be complex conjugation
real axis.
Now suppose that H is a (complex)
linear space. A (vector)
inner product
on H is a
199
-
-
map x,y-~ b , y ] from H ~ H into an admissible space Z with the following properties:
(i) (ii) (iii)
Ix, x] ~ 0, and Ix, x]
Ix,
= 0 implies x = 0
yJ = Ey, x]~;
lax I + bx2, y~ = a[xl, y~ + b[x2, yJ for complex
a and b.
When a vector inner product is defined on a space H there is a natural way in which H maybe
made into a locally convex topological vector space: a basic set of neigh-
bourhoods of the origin, IU~], is defined by
ue
= { x:
(2)
[xx]~]. I
With these ideas in mind we can make the following definitions. A
VE-space is a complex linear space on which a vector inner product is defined.
A
VH-space is a VE-space which is complete in the topology defined by equation (2).
A
LVH-space is a VH-Space for which the range space Z of the vector inner product is in fact strongly aamissible.
(These names are constructed out of the terms Vector Euclidean, Vector Hilbert,
and
Vector Hilbert with a Limit property, respectively.) One result is worth mentioning in passing, which may be described informally by saying that a VE-space can be completed (to a VH-space), uniquely up to an isomorphism. A simple example of an LVH-space is the space of all p ~ q matrices, with the usual topology,
if we define A* as the Hermitia~ adjoint of A,P as the set of all positive
semi-definite q x q matrices, and [A, B] as B*A. It is clear that in any of these spaces concepts analogous to those found useful in Hilbert space may be defined: the most important are subspaces,
operators and their
adjoints, and various special types of operators such as projection,
self-adjoint and
unitary operators. For example the adjoint of an operator T in H is an operator T ~ if one exists
such that [Tx, yS = [ x, T'y3 for all x,y in H. ( Note however that there
are no theorems guaranteeing the existence of the adjoint of an operator,
or of the
projection on to a subspace; this is the main difference from the Hilbert space
-
2 0 0
-
situation, and gives rise to certain difficulties.) For our present purpose the main interest lies in unitary operators and projections. By definition U 4 ~ unitary if U * exists and US ~ = U*U = I;
(3)
P is a projection if P ~ exists and p = p,=
It turns out that LVH-spaces have
p2.
(4)
enough structure to support a fairly rich theory of
operators, and in particular that if U is unitary then it can be represented in the form: U =
o einedE(0)'
(5)
where E(~) are projections having properties which are formally exactly the same as in Hilbert space, so that Un =
#- e i n 8dE(@)
(6)
o
f o r a l l p o s i t i v e and n e g a t i v e i n t e g e r s n .
( V E - and VH-spaces a p p e a r n o t t o have enough
structure to support this theory.) At this point the first problems suggest themselves, for (6) can be described by the statement that a (necessarily continuous) representation of the group of integers by a group of unitary operators in an LVH-space has a spectral decomposition.
In (I) it
was shown that the corresponding result is also valid for representations of the real additive group
(in other words we have obtained a natural generalisation of Stone's
theorem concerning one parameter groups of unitary operators). Problem I. Does every continuous unitary representation of a locally compact abelian group have a spectral decomposition? Problem 2.
Does every continuous positive definite function on a locally compact
abelian group with values in a strongly adm~ ssible space Z have a representation of the form J~(g)m(d~), where m is a positive Z-valued measure over the group
@f characters
In Hilbert space the answer to both questions is yes, and in fact the answer to one
-
can be deduced from the other;
201
-
it seems doubtful whether this latter statement
is valid
here. ~.
Generalised
stationar~processes
It is already clear h o w a generalised may be defined:
~Xml
Suppose
of n - m omly. Without
conclusions
not define a random variable,
variable
CI:
is a function measurable
(GSSP)
E~,X~
assumptions
representation
We shall
CI to C4 which
of the process be possible
It will of course usually be the case that a random in some sense,
but it is ~nnecessary
The set of all ramdem variables
Definition:
certain additional
but shall merely state the four conditions
definitions).
some form of integration,
if the expectation
take their values in an LVH-space H 0.
in order that a spectral
(and some necessary
stochastic process
can be drawn.
then that the random variables
must be satisfied
stationary
is a GSSP with values in a VH-space
is defined and is a function however no useful
second-order
and that the expectation
operation
is
to specify this here.
forms a complex linear space.
The set of all complex linear combinations
of "random variables"
where X and Y are H 0 - valued random variables
E X, Y
is the space of induced
random variables. C2:
Am Expectation
operator E is defined,
such that its domain is a linear subspace
of the space of induced random variables, induced ramdom variable Definition:
A second-order
X is certainly non-negative random variable
C3:
If X and Y are second-order
C~:
If I X ~ ) is a net of second-order that E [ X ~ - X~, X ~ - ~
large,
its range is contained
~ 0.
then so is X + Y.
random variables
then there exists a (second order)
and E X exists then E X
X is one for which E~X, X] is define~.
random variables,
is arbitrarily
in Z, and if am
satisfying
the Cauchy conditiom
small w h e n e v e r ~ a n d ~
random variable
are
sufficiently
X such that E ~ X -
X~, X - X ~
tends to zero. Defimition:
A null random variable X is one f o r which E~X, X ~
Definition:
Two second-order
random variables
X and Y are equivalent
if X - Y is null.
Definition:
Two second-order
random variables
X and Y are orthogonal
if ~ X ,
Concerning
these conditions
(a) Condition
and definitions,
= 0.
we may make the following
C 4 may in one sense be omitted,
since its purpose
YJ = 0.
observations.
is to ensure the
-
2 0 2 -
completeness of a certain LVH-space, and we already know that such spaces can always be completed if necessary;
nevertheless if this is necessary the additional points added to
achieve completeness do not correspond to random variables,
and hence the
interpret-
ation of results would be rather complicated. (b)
We define null random variables as above, but the later results will probably not
be useful unless these are "negligible"; in separable spaces, that E[X, X~
a suitable relationship might be, at least
= 0 implies X = 0 with probabiIity one.
A similar
requirement of usefulness will also impose further restrictions on the operator E. With these condition and definitions,
then, we define a GSSP as a process { ~ ,
with
values in HO, for which each X n is a second-order random variable such that E L ~ , is a function of n - m only. operator U which sends ~
Xm]
It follows, as in the classical case, that the linear
into ~ + 1
for all n may be regarded as a unitary operator
in the (LVH-) space of equivalence classes of random variables.
The theory sketched
in the previous section may be applied, and one is led to the following spectral representation:
Xn ~ JeinOdZ(8)
(7)
where the two sides of (7) are related by the fact that their difference is null. Again Z(e), which now takes its values in H0, has crthogonal increments in the sense that E ~ Z ( e ) ,
dZ(~)J
From this follow
= 0 if
e ~ ~.
two consequences.
(X 1 + ... + ~ ) / n ,
An ergodic theorem is valid for the averages
and if T is a continuous linear operator from the space of (equi-
valence classes of) random variables to a topological vector space, then TX n in other words TX Loire (4).
n
:
~einedTZ(S)
;
(8)
is harmonizable if we generalise in a natural way the definition of
This does not appear to have been pointed out previously even for scalar
random variables, where it leads to the conclusion that if { X n } is second-order stationary in our sense (that is with no condition
EX n
= ~eineap(e)
imposed on the mean),
then in fact
(9)
- 203 -
for some (signed)
measure H "
As an illustration
we may take p × q-matrix valued random variables,
each of whose elements introduced
is an ordinary
in the previous
section,
(scalar)
and take the expectation
then it turns out that null random variables vanishes w i t h probability and in particular tionary;
one.
random variable.
of each element
If p = 1 we are in the classical multivariate
the sequence formed by any particular
stationary.
P r o b l e m 2:
Suppose that the X n form a G a u s s i a n p r o c e s s
essentially
here in general, and ~YI Y~ •
according
but perhaps
orthogonality
now entail?
random variables tary subspace. all subspaces
belonging
of X and Y implies
independence
of EX, XS are such
that is (roughly speaking)
into the sum of its projection
via projection
such a decomposition
on the subspace
of
operators,
the case,
and the obvious
is always possible,
since not
question is
and if not what comditions
to ensure it.
Can all (scalar)
matrix-valued
is valid:
Here it is not obvious whether this is necessarily
whether
GSSP,
harmonizable
processes
and if not characterise
be represented
in the solution
as one element of a
those which can.
The solution to problem 5 does not seem to be very important, proTe useful
In the scalar case
to the infinite past and its projection on to the complemen-
are accessible
are sufficient
normal).
this is easily seen not to be the case
In the scalar case a Wold decomposition
P r o b l e m 3:
are multivariate
de-
of this type are valid.
to say that X n can be decomposed
therefore
independence:
sta -
to some suitable
It would probably be more useful to ask what scalar products
that properties Problem 4:
implies
increments
situation,
element is second-order
(for example in the matrix case if all the elements
Then what extra properties do orthogonal
separately,
by the result quoted above, but
not general
orthogonality
If we use the product
(matrices) are just those each of whose elements
if p ~ 1 then each element is harmonizable,
finition
defined as matrices
but might conceivably
of problem 7 in the next section.
4. Applications We describe
three applications,
each of which suggests
certain other questions:
another
- 204 -
problem is of course to find more applications.
None of the applications has, on the
surface, any connection with second-order stationarity. (a) Suppose we have a positive recurrent (irreducible) Marker chain with n-step
_(n)
transition probabilities ~ j equilibrium,
and stationary distribution Pi"
If the chain is in
is 1 if the and we define X n to be vector with components -(i) ±n ' where ~(i) ±n
chain is in state i ar time n and 0 otherwise, then X n is multivariate second-order stationary in the classical sense.
From this it follows easily that for each pair i,j
the sequence _(n) Pij has a Fourier representation,
a result which is due to Kendall (5).
With somewhat greater difficulty the argument can be extended to the null-recurrent case, but so far at least it has not been found possible to treat transient chains. As Feller (6) has shown that there are considerable differences between the recurrent and transient cases this is perhaps not altogether surprising. Problem 6~ Characterise the spectral m e a s u r e s ~ i j which can occur in the representation (n) Jein~i~(dS). Pij = (b)
Suppose that X n are real-valued random variables with the property that the
distribution of X n - X m depends on n and m only through their difference n - m. (This is a property similar to, but weaker than, that of stationary increments; it is in pareieXn ticular enjoyed by random walks.) Then for any given e the sequence { } is se cond-order stationary,
and it follows that there exists a random process Ze(~) such
that
eiexn
= ~ein~dZe(~).
(10)
If we write Pn for the distribution of Xn, then it is easily proved, using (10) , that the average Qn = n-1 (P1 + P2 + "'" + P n ) converges vaguely: that is to s a y f f ( x ) Q n ( d X ) converges for all f which vanish outside a compact set (depending on f).
This result
presm~m~bly does not exhaust the information contained in (10) and indeed it seems quite plausible that, subject to certain conditions on the second moments of the Xn, X n is harmonizable. -
If in fact we merely assume that ~
< co
then U N = N(exp iN -1 X n
1) second order converges to iX n as N - * @ , and it is now intuitively very appeal-
ing to suppose that ~ Xnl is harmonizable, zable processes ~ ~ n I.
being the limit of a sequence of harmoni-
- 205 -
Problem 7:
I~
Consider the class of harmonizable processes, with typical member
= [~einOdZ(e)}"
Relate convergence of a sequence of processes ~ }
behaviour of the corresponding processes ~ Z(O)) Bochner (7, c h a ~ e r
to the
and vice versa.
6) has some results connected with this problem, but they appear
not to be definitive. The situation contemplated can be considerably generalised by supposing that ~
takes
values in a (separable) locally compact topological group, and that the distribution of
&
~I
depends only on n - m.
Then if ~ U(g) ~ is any representation of the group as
a group of unitary operators in a Hilbert space U ( & ) 3, and the conclusion concerning the sequence ~
is a GSSP in the sense of section
again follows.
Here again one feels
that more information should be derivable, although in this case it is less obvious what results to expect. (c)
For a (regular) complex measure ~ in a
group, define ~
(separable) locally compact topological
by
for all Borel sets E, and the Fourier transform of p as the function ~ (U), of the members of (at least) a complete class of unitary representations of the group, where ~(U) m being Haar measure on the group.
.
(12)
Now if ~ and v are two such measures write
[~,~ the convolution of p and ~
=/U(g)m(dg)
=
~
(13)
It is easily seen that the space of measures can be
made an LVH-space in this way, the cone P in the definition of an admissible space consisting of those measures whose Fourier transform is everywhere positive-definite (positive if the Fourier transform is scalar-valued, which would be the case for abe lian groups, with the usual conventions). It is now clearly possible to define a GSSP whose terms are random measures Pn" illustration, consider the situation treated in the preceding subsection.
As an
If we write
- 206-
~n(A)
=
IA(~)
(14)
for each Bcrel set A, where IA(.) is the indicator function of A, then the sequence ~pn} is a GSSP when ~ X ~ 1 has a distribution depending on n - m only. of the ~
The convergence
follows easily, and again one feels that more information must be available,
since at least in an extended sense Xn Problem 8:
(15)
= ~ XPn(dX).
Is it possible to treat non-locally-compact
groups - for example the
additive group of a linear space - in this way? The main difficulty appears to lie in defining the cone P. Finally, two rather wild thoughts.
Suppose that we treat random variables A~(i) as in n
(a), or IA(X n) as in (14) above, where i or A are fixed, and that ~
is transient:
then there exists a random variable N such that these sequences vanish if n > N. there were no randomness present,
If
the theorem of F. and M. Riesz (see e.g. Rudin (8))
would show that the function Z(8), which woulB then be of bounded variation, is absolutely continuous: there ought, one feels, to be a corresponding result in the stochastic case, although one might hesitate to predict its exact character. whether or not ~
Moreover,
is transient, the random measure determined by Z(~) is idempotent
( in an extended sense), since ~
is its Fourier transform,
and ~
= X n.
Here again,
if no randomness were present the characterisation of idempotent measures on the circle by Helson (see e.g. Rudin (8)) would yield considerable information about Z(e), and again one is led to speculate about the existence of an analogue in this stochastic context.
- 207 -
Referenc es (1)
LOYNES, R. M. 167
(2)
Linear operators in VH-spaces, Tra~s. Amer. ~ath. Soc., 116 (1965)
-
180.
On a generalisation of second-order stationarity, Proc. Lond. Math. Soc., 15 (1965) 385 - 398.
(3)
On certain applications of the spectral representation of stationary processes, Zeits. Wahrscheimlichkeitsth. verw. Geb., 5 (1966) 180 - 186.
(@)
LOEVE, M.
Probability Theory, New York: Van Nostramd, 1955.
(5)
KENDALL, D. G.
Unitary dilations of Markov tramsition operators, and the asso-
ciated integral representations for transition probability matrices ( in Surveys in Probability and Statistics, ed. Grenander) Stockholm, 1959. (6)
FELLER, W.
.On the Fourier representation for Markov chains and the stromg ratio
theorem, (umpublished manuscript). (7)
BOCHNER. S.
Harmonic Analysis and the Theory of Probability, Berkeley: University
of California Press, 1955. (8)
RUDIN, W.
Fourier Analysis on Groups, New York: Interscience, 1962.
-
208
-
Analytical Methods in Probability Theory Eugene Lukacs (I)
O.
Introduction.
Analytical methods are used in many areas of probability theory.
This is not surprising since probability theory is a branch of analysis which utilizes methods of classical as well as modern analysis.
In the present paper we consider
certain applications of the theory of analytic functions to probability theory. In view of the great extension of the subject it is not possible to give here a complete survey.
We discuss only three topics, namely positive definiteness (section 1),
certain functional equations (section 2) and several problems which either belong to the arithmetic of distribution functions (section 3 and ~) or are related to it (section 5).
The selection of these topics is, to a certain extent, influenced by the
interests of the author. 1.
Positive definiteness.
A complex valued function of a real variable t is said to
be positive definite if (i) (ii)
f(t)
is a continuous function of t,
for any positive integer N and any real t l , t 2 , . . . , ~ and any complex
(1.1)
S
is real and non-negative.
Here
=
~k
~I'
~ 2''''' ~ N
N
N
~--
V--
j=l
k=l
f(tj-tk)
the sum
~jTk
denotes the complex conjugate to
~ k"
We note that the term non-negative definite function is sometimes used instead of positive definite function.
(1)
Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force under OFOSR grant AF-AFOSR-4~37-65.
- 209 -
Positive definite functions play a very important role in the theory of probability since it is easily seen that the Fourier-Stieltjes transforms of distribution functions are positive definite. The Fourier-Stieltjes transform of a distribution function is called the characteristic function of the distribution
(or of a corresponding random
variable)• Problems, which originally appear as questions concerning random variables or distribution functions can often be reformulated in terms of characteristic functions They lead then to functional equations which the characteristic function must satisfy, however one finds in general among the solutions of these equations also functions which are not Fourier-Stieltjes transforms of distribution functions and are therefore irrelevant in connection with the original problem• One has then to select
among the
solutions those which are characteristic functions. As a consequence of this situation two analytical problems are of great interest in certain areas of probability theory. The first is the developing of criteria which permit to decide whether a given, in general complex valued function f(t) of a real variable t is positive definite; this problem will be discussed in this section.
The second is the determination of analytical
properties of the positive definite solutions of certain differential equations and will be treated in section 2. The first problem was studied by a number of authors.
Some of these gave necessary
and sufficient conditions, others were interested primarily either in necessary or in sufficient conditions. Important work was also done concerning the positive definiteness of functions belonging to certain well defined classes.
We mention next briefly
some of these results and begin with the most important necessary and sufficient conditions. Theorem 1.1 (Bochner's theorem ~
7.
A complex valued function f(t) of a real variable
t such that f(o)=l is positive definite if, and only if, it is the Fourier-Stieltjes transform of a distribution function. A new, elegant proof of Bochner's theorem is due to Yu.V. Linnik [36], this proof is also reproduced in the book of B. V. Gnedenko E13J
Theorem 1.2
(Cram~r's criterion [4] ).
A bounded and continuous function f(t) of a
real variable t such that f(O) = 1 is positive definite if, and only if,
-
A
2 1 0
-
A
I
/
o
o
f(t-u) exp [ix(t-u)] dt du is non-negative for all real x and
all A > O. Theorem 1. 3
(wh~nchine's criterion
[l~J).
The complex valued function f(t) of a
real variable t is a characteristic function if, and only if, there exists a sequence { gn (0)~ of complex valued functions of the real variable e such that co
Ign(e) l 2
de
=I
-CO while the relation f(t)
=
lim n-~co
holds uniformely
f
gn(t+8) ~
de
-GD
in every finite t-interval.
A necessary and sufficient condition for real characteristic functions (Fourier-Stieltjes transforms of symmetric distributions) was given by
M.Mathias ~43] •
The most important of these conditions is Bochner's theorem, since it permits interesting generalizations.
The concept of positive definite functions and their relations to
measures can be extended to topological groups is available on certain algebraic structures
and a direct generalization of theorem 1.1
(see for instance W. Rudin [51] or U. Gre-
nander [147) such as locally compact abelian groups. A number of sufficient conditions for positive definiteness were also obtained. most important of these is due to G. PSlya [~8].
The
This refers to real characteristic
functions and permits the decision whether a function is positive definite by inspection of its graph. Theorem 1.4 (P61ya's condition). which is defined for real
t
Suppose further that
f(t)=O.
f(o)=l
then
f(t)
lim
Let
f(t)
be a real valued and continuous function
and is symmetric Ef(-t)=f(t)S and is convex for t > Then f(t)
is positive definite.
o.
If in addition
is the Fourier transform of an absolutely continuous distribution
function. P61ya's theorem can be generalized in the following ms~oer.
-
Theorem I. 3.
Let
f(t)
convex in the interval that
f(r)=0.
Then
211
-
be a real valued, continuous and symmetric function which is (o,r)
f(t)
and suppose that
f(t)
is positive definite.
is periodic with period 2r and
If in addition
f(o)=l
then
f(t)
is the characteristic function of a lattice distribution. Various other extensions of PSlya's theorem are known (see for instance M. Girault [11] and D. Dugu~ ~8] ). We proceed now to discuss conditions which are only necessary. Many of these are very useful if one is interested in finding positive definite solutions of differential equations, that is if one wishes to eliminate those solutions which do not meet this requirement. functions z=t+iy
The most interesting of these conditions refer to positive definite
f(t)
which agree with a regular function
in some neighbourhood of the origin (~).
A(z)
of the complex variable
It is known that such a function f(z)
is necessarily regular in a horizontal strip which has the real axis in its interior. This strip can also be the whole z-plane.
The function f(z) admits in its strip of
regularity the representation as a Fourier-Stieltjes integral. The first such result is due to P. L~vy and states that the order of an entire, positive definite function can not be inferior to one. A substantial part of the interesting necessary conditions originated in investigations concerning the characterization of the normal distribution. necessary to ~mow the conditions which a polynomial the entire function of finite order e x p ~ P ( t ~
P(t)
In this connection it is must satisfy in order that
be positive definite. The following
result is often useful. Theorem 1.6. then If
f(t)
Let
Pm(t)
a polynomial of degree
m~ 2
f(t) = exp CPm(t)];
is not positive definite.
m = 2, then it is easily seen that necessarily
real, a > 0 .
and define
For a characteristic function one has
P2(t) = -at 2 + ibt+c C
(~) i.e. one assumes that there exists a function A(z) and a positive constant 8 ~ ~ such that
=
with
a,b
Oe
, regular in the circle I z I <9,
A(t) = f(t)
for
It I ~ .
-
2 1 2
-
Theorem 1.6 is a particular case of a more general proposition. Theorem 1. 7 (~arcinkiewicz F42]). ponent of convergence
~1
An entire function of finite order
is less than
~
~ >o
whose ex-
can not be a characteristic function.
In the statistical literature one refers often to theorem 1.6 as the theorem of Marcinkiewicz. m Theorem 1.8.
Let
=
Then
en~P(t)] .
P(t) = ~ fn(t)
cktk
be a polynomial of degree m > 2 and write
is not positive definite. Here
ponential function defined by the relations k = 1,2,...
el(x ) = ex
en(X) ,
ek(x)
fn(t) =
is the iterated ex =
exp~ek_l(X) ~
for
.
Further extensions were given by several authors. functions of the form
f(t) = g ( t ) e n E P m ( t ~
having certain properties while polynomial of degree (i)
m .
where
en~Pm(t) ]
I. F. Christensen F3~ considered g(t)
is a characteristic function
is again the iterated exponential of a
Several cases were studied
g(t)
is an entire characteristic function of a lattice distri-
bution which has the origin as a lattice point. ticular case ~1 > o,
~2 > c
Here the par-
g(t) = exp [~l(e it - 1) + ~2(e -it - 1)]
with
is of interest in connection with certain cha-
racterization problems. (ii)
g(t)
is the characteristic function of a discrete distribution
which has only a finite number of discontinuity points. A necessary condition for the positive definiteness of the polynomial
Pm(t)
should not exceed 2.
polynomial of degree
where m >2
f(t)
and g(t)
showed that under these conditions
is that the degree
of
considered functions of the form
is a non-constant entire function,
P(t)
is an entire function of order ~ ~ m . f(t)
m
Cairoli ~2] gave similar results for mero-
morphic functions of finite order, H. D. Miller ~ ] h(t) = g(t)f~exp(P(t)]
f(t)
is a He
can not be positive definite.
Yu. V. Linnik asked (problem 1 in section 13.2 of his book ~36~--this section is devo-
213
-
voted to umsolved
problems)
whether
exp~g(t)]
entire function of order I and minimal
-
can be positive
type.
Theorem I. ~. tion
if
g(t)
is an
The problem was answered in the negative
by I. V. 0strovskii ~45~ who obtained the following only indications
definite
result in a note which contains
of his method but not a complete proof.
(0strovskii)
If the entire transcendental for
Re f(x+iy)~ M ( l y l )
- ~
~ x, y ~ + ~
function
where
satisfies
the rela-
M(r) = max i f(z) I Iz1=r
then
lim sup r-llog M(r)~ 0. r~oo As a consequence, entire function
a function
of the form
of order 1 and minimal
I can not be positive
on two symmetrically
definite functions val.
fumctions
suggests
which are positive
ty of their extension. function
f(t)
holds for any
extension function
.
of a function
located intervals.
definite
on the interval
~I'''''
defined
which assure the uniqueness
t
.
is positive
of
I t~La
such that
problem to study the to a positive
on the interval
definite
studies
definite The fact
of conditions of these prob-
•
is the determination
of the complex variable f(t)
Iti~a •
the investigation
Very interesting
[207 F20a~
definite whenever
(I .1)
This means that one has to obtain con-
of the extension.
problem was obtained by C. S. Herz
t1,...,~
of at least one function which is positive
Another question which is of some interest G(z)
if the relation
analytical
and positive defimite for
lems were carried out b y M. G. Krein
G~f(t~
the concept
and to study the possibili-
tti~a
is in general not unique motivates
plex valued fumctions
then that positive
to introduce
and all real
N
and agrees with the given function
that this extension
of order inferior to
if they are given on a finite inter-
It is then a very interesting
ditions which assure the existence t
determined
It follows
definite on a finite interval
defined for all real values of
for all real
is either an
definite functions which agree every-
that it might be desirable
and amy complex
Itjl
g(t)
In obvious analogy to the earlier definition we say that the
is positive N
type or an entire function
two positive
are not necessarily
This situation
where
definite.
Theorem 1.4 cam be used to construct where except
h(t) = exp~g(t~
z
of the class of com-
which have the property
is positive
definite.
[15] who showed that necessarily
A solution
that of this
-
G(z) = >--
214
-
am, n Zm ~n
mtn=o with a
real and non-negative. He obtained the solution in a more general framework m,n by considering positive definite functions on locally compact Abelian groups. 2.
Certain functional e~uations.
A problem which we mentioned already briefly deals
with the positive definite solutions of certain differential equations. ting result is due to A. A. Zinger and Yu. V. Linnik ~58] [29].
A very interes-
We consider an ordi-
nary differential equation
(2.1)
~[Ajl
• "''~n
l•-(J1+'''+Jn) f(Jq) (t)...f (in) (t) = cEf(t)] n
where the Aj1 "''in are real constants and where the s~mmation runs over all non-negative integers
jl,...,jn
which satisfy the inequality
(2.2)
Here
jl+...+jn
p
_~ p .
is an integer such that at least one coefficient
is different from zero.
I
AJ1"''~n
with
jl+...+jn=p
We denote the order of this differential equation by
m .
We
introduce the polynomial
(2.3)
A(xl,...,x n)
where the first Sl~mmation~ -~ n
integers
(1,...,n)
satisfying (2.2) .
I = n-T.~
Jl Ajl...j n Xkl
Jn ... Xkn
is taken over all permutations
(kl,...,kn)
while the second sm~mm~tion is taken over all
The differential equation (2.2)
if the polynomial (2.3) is non-negative.
of the first
(jl,...,jn)
is said to be positive definite
We state now the results of A. A. Zinger
and Yu. V. Linnik. Theorem 2.1.
All positive definite functions which are in a neighborhood of the ori-
gin solutions of a positive definite differential orders at the origin.
equation have derivatives of a l l
-
Theorem 2.2.
215
-
Suppose that the positive definite function is, in a neighborhood of the
origin a solution of the positive definite differential equation (2.1) and assume that man-1.
Then
f(t)
is am entire function.
The study which led to these results was motivated by the wish to obtain solutions of certain regression problems which occur in connection with the characterization of the normal distribution.
In fact, Zinger and Limmik gave in their paper ~58] further conditions
on the polynomial (2.3) which yield such a characterization.
However,
the assumptions of theorem 2.2 are too restrictive in this connection.
it seems that It would be
highly desirable to obtain additional theorems describing the analytical properties of differential equations similar to (2.1) but using a different set of assumptions. Several characterizations 6f the Gsmma distribution are known which lead also to differential equations of a similar form.
For a suitable choice of the parameters the
Gamma distribution is a rational function.
It would be interesting to find conditions
w h i c h assure that the positive definite solutions of these differential equations are rational functions. Very interesting work on a functional equation and related problems is comnected with the equation defining the stable dintributions.
It would be beyond the scope of this
p a p e r to discuss the probabilistic meaning of these studies and we describe therefore only the analytical aspect of the problem and state it in the a l l
characteristic fumctions which have property that to every
a b >o
following way: Determime bl>O , b2~o
corresponds
such that the functional equation
(2.4)
f(blt)
holds for all real
t
f(b2t)
and some real ~
=
f(bt)
ei~
.
The solution is well known, the only characteristic functions which satisfy this equation are given by
log f(t)
(2.5)
where
c,~,~
= i a t - o
are real constants such that
Iti,
c~o,
)}
l~l&l , 0
while
a is a real
-
number.
o~(ltl,A)
The function
~
(Itl,&)
-
is defined by I
(2.5a)
2 1 6
tan-~-
if
d.~l
2 ~--
if
06 =I
:
log I t I
The functions defined by (2.5) and (2.5a) are called stable characteristic functions. They are the only positive definite solutions of (2.4) which do not vanish identically. The parameter
@( is called the exponent of the stable distribution.
The characteris-
tic functions (2.5) are absolutely integrable, the corresponding frequency (density) functions are therefore absolutely continuous and are given by
f
--O0
Explicit expressions for these densities are only known in a few isolated cases, it is therefore of interest to study their properties and we mention next some of the results obtained by using analytical methods. It is not difficult to show that the frequency function corresponding to a stable characteristic function with exponent ~ A 1
is regular; f o r ~ 1
it is an entire
function while for % =1 the radius of convergence of its power series in the neighborhood of a point of the real axis is at least equal to plicated i f ~ <
(2.6)
1.
p(x)
61
functions
The situation is more com-
In this case Skorohod ~54] succeeded in showing that
=
f x_l+1(x-~) Ixl-I~2( Ixl -~)
where
c.
and
~2
are entire functions.
41
and ~ 2
for
x>o
for
x~o
It would be interesting to know whether the
are entire functions of finite order.
In this case it would
be desirable to study the dependence of the order on the parameters a,~,~,c.
The
asymptotic behaviour of stable frequency functions was thoroughly investigated.
We
refer here to the papers by A. V. Skorohod[ 55] and V. M. Zolotarev [61], the last author [ 60S expressed also the frequency function of a stable characteristic function I with exponent ~ > 1 in terms of a density with exponent ~- and a suitably modified
-
parameter ~ .
217
-
V. M. Zolotarev [62J obtained also a representation of the distribution
functions corresponding to stable characteristic functions by an integral over a finite interval.
The interval of integration, as well as the integrand depend on the parame-
ters of the stable distributions. We list finally a few functional equations which were studied in connection with certain probabilistic questions, mostly in connection with characterization problems. (a)
D. A. Raikov C49] studied the functional equation
(2.7)
where
f(t)
a,c I , and
c2
=
e iat f(clt) f(c2t)
are real, ci>o , c2>o .
He showed that the only positive defi-
nite solution which is not identically zero is the characteristic function of the normal distribution. (b)
The functional equation involving
n
functions
n
(2.8)
77
S=I
f1(t),...,fn(t)
n
fj(aju+bjv)
=
.
~
S=1 fs(aju)
fs(bsv)
was studied by many authors in connection with the oroblem of independently distributed linear forms in independent random variables. (@) Assuming ajbj = 0 and fj(o) = 1 (j=1,...,n) it is possible to show that the only positive definite solutions fj(t) of (2.8) are functions fj(t) which are characteristic functions of normal distributions (possibly with different parameters). (c) J. Marcinkiewicz E42~ studied the functional equation
(2.9)
-KU- f(ajt) S
=
-~T
f(bjt)
.
The product can here be finite or infinite (in the latter case it is assumed to be uniformly convergent in every finite interval); the
(*) The final solution, removing all ~ e c e s s a r y
{ aj 3
and
{ bj }
are sequences
assumptions made by earlier authors
is due to G. Darmois [6] F7] and V. P. Skitovich [52] [ 53] who obtained it independently at about the same time.
-
of real numbers such that the permutation.
~ I ajlJ
2 1 8
-
can not be obtained from the
Marcimkiewicz supposed that
f(t)
~ } bj 17 by a
has derivatives of all orders
amd showed that the only positive definite solution of this equation is the characteristic function of the normal distribution.
Marcimkiewicz derived theorems 1.6 and 1.7
in connection with the investigation of equation (2.9).
These theorems play a crucial
role in his proof. (d)
A similar problem led to the equation
TT
(2.10)
f(ajt)
:
f(t)
J where one had only to assume that tions.
T
aj2 ~ 1
but needed no differentiability assump-
R. G. Laha-E. Lukacs [25~ showed that the characteristic function satisfying
(2.10) is the characteristic function of the normal distribution. The positive definite solutions of the functional equations mentioned above have the form
exp~P(t)~
where
P(t)
is a polynomial of the second degree.
The functional
equations (2.7), (2.8), (2.9) and (2.10) were motivated by probabilistic problems. Modifications of the probabilistic assumptions lead to different equations whose positive definite solutions have a different form. These equations occur in connection with characterization problems (see [38J, [ ~0], [ 28]). These studies stimulated also some work on functional equations which do not insist on the positive definiteness of the solutions (see f . r . E .
~.
Vincze [57], Laha-Lukacs-R~nyi [24J).
Multiple factorizations.
It is well known that the product of two characteristic
functions is always a characteristic function. teristic function and
f2(t)
=
f(t)
It is possible to write every charac-
as the product of two characteristic functions
e-imt (m real).
The product representation of a characteristic function
is said to be trivial if one of the factors has the form
e imt .
The arithmetic of
distribution functions deals with the non-trivial product representations tions, factorizations)
fl(t)=f(t) e imt
(decomposi-
of characteristic functions into factors which are themselves
characteristic fumctions.
It is easily seen that there exist indecomposable charac-
-
219
-
teristic functions (*) . The indecomposable characteristic functions play, to a certain extent, a role similar to that of the prime numbers in ordinary arithmetic.
However,
there are several very important differences. We mention here only two peculiarities of the arithmetic of distribution functions which indicate that the analogy with ordinary arithmetic does not go very far. (i) There exists no unique decomposition into prime factors. For example, the characteristic function f(t) = ~ j ~ o
eitj admits two different decompositions
(see
~pg.78).
(ii) The cancellation law does not hold; this means that it is possible that a characteristic function f(t) admits two decompositions of the form (3.1)
f(t) = f1(t)
f2(t) = f1(t)
f3(t)
where f2(t) = f3(t). The first example of this kind is due to B. V. Gnedenko
12 .
Using theorem 1.5 one can construct characteristic functions f1(t) and f3(t) such that f1(t) ~ f3(t) for
Itl~r (r>o) while f1(t) ~ 0 for
(5.1 a)
f(t) = [ f 1 ( t ~ 2
=
Itl>r and f3(t) $ 0 for
f1(t ) f3(t ) .
It is known that the quotient of two characteristic
functions is, in general, not a
characteristic function. However, we see from formulae
(3.1) and (3.1a) that the quotient
of two characteristic functions is not uniquely defined, characteristic functions.
Itlmr. We have then
even in cases where it is a
Pormula (3.1) indicates also the intimate connection which
exists between the unique determination of a quotient and the non-vanishing of a characteristic function. We bring next some results which are due to T. Kawata
~8~
but are not
sufficiently well known, probably because they were published during the last war. Theorem 3.2 deals with the possibility of constructing characteristic functions admitting decompositions of the form (3.1a). The next theorem gives conditions which permit the construction of characteristic functions vanishing outside a fixed interval;
the existe of such functions
is needed in our construction.
(*)
It can be shown that every characteristic function which belongs to a purely discrete
distribution with exactly two discontinuity points is indecomposable. to construct continuous indecomposable distributions.
It is also possible
220
Theorem ~.1
Let
e(u)
-
be a positive, non-decreasing function defined in
(0,+oo) such
that OO
1 and let
b
be an arbitrary (but fixed) positive number.
tion function
F(x)
which satisfies for every
(3-3)
F(-x+a) - F ( - x - a )
and whose characteristic funtion
f(t)
a
Then there exists a distribu-
the relation
= ~ [ exp[--e(x)]S vanishes for
(as x ~ )
Itl~b .
For the proof one needs a result due to N. Levinson [26] [ 2 6 ~ Lemma ~.1 and let
Let b
function
e(u)
•
be a positive, non-decreasing function which satisfies (3.2)
be an arbitrary but fixed positive number. G(x)
and A. I. Ingham [17]
Then there exists a non-null
such that
(3.4)
G(x)
= ~ {exp[-~(Ixl)]}
(aslxl~oo)
which has the property that its Fourier transform OO
g(u)
vanishes for
1
J
G(x) e-iUXdx
luI~b •
To prove theorem 3.1 we consider
e(2u)
which has the same properties as
apply lemma 3.1 replacing
b/2 .
We put
f(t)
b
by
=
l y
i
g(x)
e(u)
and
g-~x
OO
where f(t)
A = _~
Ig(x)12dx .
According t o a c l a s s i c a l
is a characteristic function such that
result
f(t) = 0
for
of K h i u c h i n e ( t h e o r e m 1 . 3 ) Itl~b .
Using the inversion formula, Parseval's theorem and relation (3.~) one sees by means of a simple computation that condition (3.3) is satisfied.
-
Theorem ~.2.
Let
e(u)
-
be a positive, non-decreasing function which satisfies (3.2).
Then there exists a distribution function a decomposition of the form (3.1a) • all positive
221
F(x)
whose characteristic function aam~ts
~oreover,
F(x)
satisfies condition (3.3) for
a .
We consider again
O(2u)
e(u)
instead of
exists a distribution function
Fl(X)
and take
b= ~/2
in theorem 3.1, then there
whose characteristic function
fl(t)
has the
property that (3.5)
fl(t) = 0
for
Itl>~
This function is obtained by means of the function
g(u)
of lemm~ 3.1 and by defining
--CO
--CO
we note also that (3.7)
g(x) = o
Ixl>
for
and that Fl(X) satisfies the relation
FI(-X
+
a) - FI(-Z- a) = O{e:r9 [-O(2:x~]
(as x -* co )
In view of (3.6) and (3-7) one sees that fl(t) -~We introduce now a function f2(t) which is periodic with period 2 ~ r a n d which coincides with fl(t) for Itl~TC. We see from (3.6) and (3.7) that fl(Tg) that f2(t) is continuous. Let
1
cn
=
fl(-~)
/f2Ct)
1
e-int dt
qT
~-~
~ - ~
0
so
be the sequence of Fourier coefficients of f2(t),then
fe-int
I" / 7T
1
=
~+x g(x) e
~~y
[
/
-~+x
g--~
•
e -Iny dy] dx
]
-
222
-
It follows from (3.7) that
Cn
=
y
~
g(x) e inx
e-lny
/
-g
~7
-7
2 I
=
J
e imx
g(x)
(3.8a)
~-
0
and we see from the relation
f2(o)
cn=
n=-cO
=
fl(o)
=I
that GD
(3.8b) Z:~--GO
On
=
so that f2(t)
It followB that
f2(t)
=
~-c n e -int n=-oo
is the characteristic
function of a lattice distribution whose
discontinuity points are contained in the set of integers. bution at the point cal with
f1(t)
n
equals c n (n = o, ~I, ~2, ...). Clearly
f2(t)
is not identi-
and one has
f(t) Thus f(t)
The saltus of this distri-
=
[fl(t)]
admits a decomposition
function which corresponds to f(t) tained by a somewhat
2
=
f1(t ) f2(t ) •
of the form (3.1a). Let F(x) be the distribution ; the statement
lengthy but straightforward
that
F(x)
satisfies (3.3) is ob-
computation.
T.Kawata obtained also a sufficient condition which assures the impossibility of a decomposition (3.1a). We mention next this result without giving its proof. Theorem 3.3
Let F(x)
be a distribution function and let
e(u) be a positive,
non-
- 223 -
-decreasing function defined on (0, +oo) such that co
I
If for some a > o F(x)
u
the relation (3.3) holds and if the characteristic function
f(t)
oz
aam~ ts the decomposition
f(t) then f2(t)
= fl(t)
f2(t)
is uniquely determined by
f(t)
and
f1(t).
4. Factorization of infinitely divisible distributipns. A very important chapter of the arithmetic of distribution functions deals with infinitely divisible characteristic functions. A classical theorem of Khinchine (see [39U Pg. 115 )
asserts that a chmracteristic function which has no indecomposable factors is
infinitely divisible. The converse statement is not true, P.L~vy [27] constructed a characteristic function which is a product of three Poisson type characteristic functions. This function is therefore infinitely divisible, however it admits a second decomposition into two indecomposable factors. Thus one has the rather surprising result that the product of two indecomposable characteristic functions can be infinitely divisible. Paul L~vy's construction is based on a very interesting study of functions of the form exp[P(t~
where
P(t)
is a polynomial. It is easily seen that even if the
polynomial P(t)
has some negative coefficients exp [P(t)~
expansion about
t = o
has only non-negative coefficients. P.L~vy has obtained ne-
cessary and sufficient conditions which the polynomial that e x p [ P ( t ) ]
can be a function whose
should have only
P(t)
must satisfy in order
non-negative coefficients. This is a result of in-
dependent ~ a l y t i c a l interest which has important consequences in the arithmetic of distribution functions. Similar results were also obtained by D.A.Raikov, a brief survey of these may be found in [39~ • The situation which we just described suggests the problem of studying the class I o of infinitely divisible characteristic functions which have no indecomposable components. This class contains certainly the Normal and the Poisson ditributions; this follows from two famous theorems: The theorem of Cram~r which asserts that the factors of a
- 224 -
Normal distribution are all normal and the theorem of Raikov which makes a similar statement concerning the Poisson distribution.
In the papers of P. L~vy ~27] and D. A.
Raikov ~49~ we find a few additional examples of characteristic functions belonging to Io;
D. A. Raikov and P. L @ v y w e r e
investigating in these papers the multiplicatlve
structure of finite convolutions of Poisson distributions. one realizes that the study of the structure of the class
Looking at these results Io
central problems of the arithmetic of distribution functions.
constitutes one of the The problem was already
formulated by Raikov in [49] but the first significant results were obtained about 20 years later when Yu. V. Linnik published a number of very important papers [31] [32] ~3]
~34]
and a monograph ~36~.
The starting point of Linnik's investigations was a
generalization of the theorems of Cram~r and Raikov (Linnik [31~; the analytical methods developed in the derivation of this result suggested the approach which he used in his subsequent analysis of the structure of the class
Theorem 4.1 (Linnik).
Let
I o.
We state next this first result.
f(t) = exp{Z(eit-1) + i p t - ~ t 2 / 2 ~
(p real ,~2 & o,~ ~ o)
be the characteristic function of the convolution of a Normal and of a Poisson distribution.
Suppose that
f(t)
admits a decomposition
= e x p ~ j ( e it - I) + iHjt -~2t2/2]
(j=I,2)
where
f(t) = fl(t) f2(t).
Then
~=~I+~2, ~2=~12 + G 2 2, Aj-~o ,g~ ~ o.
Theorem 4.1 contains Cram~r's theorem and Raikov's theorem as special cases. the method of proof requires more powerful analytical tools. fact that Cram~r's
fj(t) =
However,
This is explained by the
theorem aeals with an entire function of finite order while Raikov's
theorem deals with a periodic characteristic function with a real period.
The assump-
tions of theorem 4.1 discard both these advantages so that the proof becomes much more complicated. Recently I. V. 0strovskii [46~ gave a simple proof of a theorem from which theorem 4.1 follows immediately.
We state next this result and give a brief indication of its proof.
We use a terminology introduced by Linnik and say that an entire function ¢ (t) complex variable
t=g+iw
(g,T real) is a ridge function if
of the
- 225 -
If
Theorem ~.2.
~l(t)
and ¢2(t) are entire ridge functions with
~1(0) = ¢2(0) = 1
and if
¢l(t)~2(t)
(4.1) and
where 2 ~0,~-~0
~
= expI~(eit-1 )
- ~2+i~t~
are real, then
iflktj
¢ k(t) = exp[~k(eit-1) - ykt2+ where
~k~O, ~k*O
and
&
(~1,2)
are real.
We list next several results which are needed for the proof of theorem 4.2.
Some of
these are from the theory of analytic characteristic functions, some from the theory of functions of a complex variable. Statement (A).
All entire characteristic functions are ridge functions.
Statement (B).
All factors of entire characteristic functions are entire characteris-
tic functions. It follows immediately from statements (A) and (B) that theorem 4.2 implies theorem 4.1. Statement (C). ~_ - M S I~I
I f ~(t)
is an entire ridge function such that ~(0)=1
(-co <~
Statement (D). (~)
Let
Izl~co ,~ @arg z ~
F(z)
U@>0
and which satisfies the conditions
If(z)l
(ii)
@~ exp [ I z l g ]
If(xei~)l lf(z)l
then
-~ K eLlzl
does not depend on ~ .
be a function which is analytic in the angle ~ = ~zI0
(i)
Statement (E~. for
then log ~(i~)
Let
f(m)
for all
=
K
and
}-
ck
To prove theorem 4.2 we note that
~ 6 J pg. 393
(~1 (t)
L
zg~l. T
such that
are positive constants.
Ta < z k
and (~2(t)
J
p~z/~-~)
(MI>O, ~ 0 )
I f(xei~)l
~ ~2
k= - ~
zeros. Therefore (~) See E. Hille
z g~l where
be an entire function with period
0~_IzI~oo where
f(z)
~2;
for
where
~
=
I f(z)l
Then
[JTIL(2x~-!].
are entire functions without
- 226 -
(4.2)
(k=1,a)
Ck(t) : expI~k(t) }
where the VYk(t)
are entire functions which are real for purely imaginary values of the
argument and which are zero for (4.3)
t=o.
g(z) :
Clearly,
g(z)
is real for real
(4.4)
z
g(o) =
Let ~1(iz). and O.
Theorem 4.2 is proven if one shows that (#) where
g(z> : ~1' ~I
and ~1
~l(e z - 1) + ~1 z2 + ~ 1 z
are real, ~1 @ O, ~1 ~ O.
We introduce the function (4.5)
u(x,y)
=
(x,y real)
Re g(x+iy)
For the proof of the theorem one needs several lemm~s. Lemma 4.1.
For real
x
and
y
the inequality
3 R e x + ~ x 2 + y 2) + ~(Ixl)
lu(x,y)I
holds. To obtain this relation one shows first that (4.6) 0 ~ u(x,o)-u(x,y) @ 2~eXsin 2 y/2 + f 2 and then (4.7)
lu(x,o)l ~ ~e x + y x 2 + ~(Ixl).
Relation (4.6) follows easily from (4.1), (4.2) and the ridge property. (4.7) is obtained by means of statement (C) and the equation which is a consequence of (4.1) and (4.2).
The estimate
~1(ir)=ke--~+~-~r-~2(iT)
Combining (4.6) and (4.7) one obtains the
statement of the lemma. Lemma 4.q and the formula of Schwarz 2~
(4.8)
g(z+~) =
~I
Z 0
u(x+cos~ ,y+sin~ )
~ei~+~ eiW_
d~
+ i
Im g(z)
- 227 -
where
z=x + iy
and
I~
(4.9)
~ I
g(z) =
Since the entire fumction
=
oo ~--
ak zk
y
that of
g(z)
a k.
z
it aamits the expansion
Therefore we have for real
fixed, the right side of (4.10) is am entire function of can also be defined for complex values of
x
g(z) =
and
x
x, we see therefore
and is an entire function
Formula (4.9) yields then the estimate
(4.11)
u(x,y) =
k(x) = u(x,o) - u(x,2~);
Let
is real for real
u(x,y) = ~ {g(x+iy) + g(x-iy)}
u(x,y) x.
( I z l *oo)
~ (Izle x + Izl 3 )
with real coefficients
(4.1o)
For
yield then the estimate
O(ixleRe x + ix13)
(Ix~m).
this is an entire fumction of the complex variable
x.
The following lemma is crucial for the proof of theorem 4.2. Lemma 4.2.
The fumction
k(x)
is a constamt.
U s i n g (4.6) we see that
(4.1a)
k(x) = O(I)
if
Im x = o
amd from (~.11) that (4.13)
k(x) =
0(Ix13)
as
Ixl-~oo
if
Re
x=o
amd have also
(4.14)
k(x) = 0(explxl 3/2)
for all complex
x.
Let (4.15)
e(x) =
k(x) (x+l)-3
This function is regular in the half-plame WT / arg x ~ 0 ~_~=
amd
0 • arg x ~ ~
Re x ~_ 0
aud satisfies in each angle
the conditions of statement
~~r- . It follows then from statement (D) that
@(x)=O (I)
(D)
for Re x~o.
with
~ = ~-, 2
-
Hence, for
-
Re x ~ 0
(@.16)
k(x) =
£ ( I x l 3)
In a similar manner we use the function is also valid for that
228
Re x ~ 0. Therefore
k(x) is a polynomial
as
el(X) = k(x)
(x-l) -3
k(x) = £(Ixl 3) for
of degree not exceeding
3.
Ixl~oo. to show that (@.16)
l~eo
and we conclude
This is compatible with the
estimate (4.12) only if k(x) is constam$. It follows from (4.10) that -2k(z) = g(z+2~rl) + g(z-2~i) - 2g(z) amd we see from lemma @.2 that (4.17)
where
g(z)
g(z+2~i)
o
is a comstamt.
(4.18)
satisfies the difference equation
+ g(z-2~i)
- 2g(z)
=
c
Let g1(z)
we conclude from (@.17) that
= g(z) - g ( z - 2 ~ i )
g1(z)
cz -~-T
is periodic with period
21~ Moreover we see from
(@.9) that
(@.19)
gl(z)
We apply statement
(E)
=
0{exp(~ Iz,)~
and get ( s i m c e ~ = l )
gl(z) = A o + A1 e÷z + A2 e-z
3 where
Ao,A 1,A2are constants.
x.-oo .
This is therefore
(@.20)
According to (@.9)
also true for
g(x) = O(Ixl
gl(x), hence
) for
x
A 2 = 0 and
(Bo,B 1 ,B2 constant).
g(z) - g(z-2~i) = Bo+ B lz + B2eZ
We imtroduce B (@.21)
g2(z) = g(z)
+ iB 1 o
z 2~i
We see them from (@.20) that
real and
B1 -~i
z2
B2 2~i
ez
- 229 -
s2(z) - ~ 2 ( ~ i )
( c constant)
= 2B o + ±BI(I+~) = c
.
cz
Let
g3 ( z ) = g2 ( z ) - ' Z ~
It is then easily seen that le to apply statement (E).
g3(z)
is periodic with period
2~i
so that it is possib-
Repeating the reasoning which led to (3.20) we see that
= Co+0 le z
~2(z)
(Co,C 1 constants)
and obtain, using (~.21) the relation
g(z)
(~.22)
where
Do,DI,D2,D3,D ~
(~.22a)
DO+DIZ+D2z2 + D3eZ + Dq.zez Since
are constants.
clude that the coefficients
Prom (~.5)
=
are real.
Do,...,D @
DO
g(z)
= - D3
is real for real
z
one can con-
It follows from (@.@) that
•
and (~.7) we conclude that
(~.~b)
V4
0
=
Moreover,
u(x,o) - u(x,y) =
In view of (@.6) this is non-neEative
that
for all real
D2y2+2D 3 ex sin 2
for all real
D2~ 2 ÷ 2 D 3 e X
x.
We l e t
x
and
•
y.
We put
y=~r and see
@ 0
first tend to + co, then to - co and see that necessarily
(~.22c)
D3
.~ 0
(~.22d)
D2
@
We write
x
~
0 and obtain from (@.22),
(@.222) and (@.22b)
230 -
(#.23)
g(z) = % ( e z - 1 )
+ ~Iz 2 + /~z
.
This is equation (*) ; it is seen from (4.22c) and (4.22d) that the coefficients
conditions on the
~1 and Z1 are satisfied so that ~he theorem 4.2, and therefore also theorem
4.1, are proven. Theorem 4.1 was the starting point for Linnik's investigations ([32~,[33], [34], [36]) on th# structure of the class
I o.
We mention here only two of his subsequent results.
To formulate these it is convenient to introduce the following terminology. An infinitely
divisible characteristic function
f(t)
is said to belong to the class
if it admits a representation of the form 2
in f ( t ) =
~t-
~2
+
where the coefficients m, r_~O(r=l,2, ;
co
z~=lm~= - ~ ~m,r
(e itpm'r -1-
~ ,~,Im, r,Pm, r
m=0,+I,+2,...)
~
)
1+Pro,r are real and # _aO,
while ~ m , l > O
and~
m,240
(re=O,+1 ,+2,... )
and where the conditions 2 2
(b)
"1+ 2
-1 the numbers ~ m + l , r ~ m , r
(r=-1,2; m=0,~1,~2,...)
are positive integers different from 1 are satisfied. Theorem 4.~.
If an infinitely divisible characteristic function
component (270) Theorem 4.4.
belongs to
with normal
f(t)
belongs to ~0
I o then i~ belongs to the class ~D •
If an infinitely divisible characteristic function
and if there exist constants
f(t)
c>o,~o
and
mo
such that
-
(4.2~,)
;& m,r
for m~ m o and r = 1,2,
~ exp f - exp (c
231
-
IPm,r 1 Z+a)}
then f(t) belongs to I o.
Yu.V.Linnik (*) conjectured that condition (@.2~) could be weakened.
I.V. 0strovskil
~47~ succeeded recently in proving the correctness of Linnik's conjecture and replaced (@.24) by the requirement that 2
(4.25)
/~ m,r = O { e x p ( - kpm,r )t
as m -->co, r = 1, 2,
should be satisfied for some k > O.
5- Analytical problems related to the arithmetic of distribution functions. The results discussed in this section are strictly speaking of an analytical nature. However, they were motivated by the arithmetic of distribution functions, notably by the theorems of Cram@r and Raikov. While the problems treated in this section are closely related to the arithmetic of distribution functions, it is nevertheless not possible to interpret these results probabilistically and to reformulate them in terms of random variables. We mention first an extension of Cram@r~ theorem to the Fourier transforms of functions of bounded variation; this result is ame to Yu. V. Linnik-V. P.Skitovich [ 37] Theorem ~.1.
Let ~
(-co, +co) such that
be the class of functions _~
symmetric functions o f ~
(5.1)
/
V(x)
of bounded variation (~) on
dV(x) = 1. Suppose further that V1(x) and that for some ~ >
IdVj(x) l
=
OEexg(-yl+~)]
•
and
V2(x)
are
0 the estimate
holds f o r
j = 1, 2.
]xl>y co If the convolution
V1 * V2
=
/ -co
V 1(x-y) dV2(Y)
is normal then
V1
and V 2 are
normal distribution functions .
(~) (~)
see D ~ ' Pg" 236 ~ 8). We note that all distribution functions belong to ~
. We say that a function
V(x) of bounded variation is s~maetric if V(x)=1-V(-x) for all continuity points of V(x). We denote by Ix~;yl dV(x)I
the total variation of V(x) over the set [~xl> y]
.
232
-
-
We note that condition (5.1) is essential, this was shown by Linnik and Skitovich by means of an example. The statement of theorem 5.fl can be reformulated in terms of "generalized ramdom variables " defined on a suitable measure space.
Using this terminology it is also possible
to prove a similar extension of the Skitovich-Darmois theorem.
This extension, as well
as the example mentioned may be found in Linmik's monograph [36~ (see section 6.~). Cram~r's theorem was extended by A. A. Zimger-Yu. V. Linnik [59~ in a different direction by proving the following theorem. Theorem ~.2.
Let fl(t), f2(t),...,fs(t)
~1,d2,...,~s
be arbitrary positive real numbers.
~[f~(t)] j=l "
(5.2)
6(~
holds in a neighborhood of the point (j=l,2,...,s)
be arbitrary characteristic functions and let
:
Suppose that the relation
exp[ipt - ~2t2/23
t=o.
Then the characteristic fumctions
fj(t)
belong to normal distributions.
The theorem follows immediately from Cram@r's theorem if either all the nal numbers or if the
fj(t)
~j
are ratio-
(j=l,...,s) are assumed to be ~nfimitely divisible. The
proof of theorem 5.2 in full generality is however more complicated than the proof of Cram@r's theorem. A number of similar results (~) were obtaimed by several authors
(R. G. Laha[21],
Zinger-Linnik [ 59], Dugu~ [ 9], [ 10], Teicher [56], Laha-Lukacs [ 23], L i m n i k [ 36] (pg
78 )) who studied a relation of more general form than (5.2), namely,
s q-F j=l
(5.3)
~j [fj(t)]
= f(t)
where the fj(t) are again arbitrary characteristic functions, the ~ numbers while
f(t)
is a characteristic function which belongs to a certaim f~m~ly~-.
The theorems assert then that each of the sometimes described by the statement that
fj(t)
belongs also to ~-
Often called
~-
. This property is
~- is strongly factor closed. We list a few
~tron~l~ factor closed families of characteristic functions: (~)
positive real
decomposition or ~ -
(I) The Normal f~m41y;
factorization theorems.
- 233-
(2) The Poisson family; functioms;
(3) Analytic characteristic functions;
(5) Entire characteristic functions of finite order;
tions which have moments up to order2m; tributions;
(4) Entire characteristic (6) Characteristic func-
(7) Characteristic functions of binomial dis-
(8) Lattice distributions.
Condition (5.2) of theorem 5.2 can be weakened by requiring that this relation should hold only for a sequence Yu. V. L i n n i k ~ 3 5 ]
~ tk} of real numbers such that
obtained also an
lim tk = 0 • k~¢o
~-factorization theorem for certain infinitely
divisible characteristic functions. He also suggested the problem of modifying the conditions of t h e o r e m 5 . 2
by putting an infinite product
q-f[ fj(t)] j=l on the left hand side of formula (5.2). This problem was solved by L. V. Mamay[41]; work in this direction was continued b y R . G . L a h a ~22~, B.Ramachandran [50land R.Cuppens
[5]. We finally mention a result due to Yu. V. Linnlk~37~ Skitoviah
[37~
(see also F363 Pg. 98 and L i ~ k -
who showed that a particular case of the Skitovich-Darmois theorem is
equivalent to Cram~r's theorem while the general theorem of Skitovich-Darmois is a consequence of theorem 5.2.
- 234 -
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Trudi mat.
- 239 -
Martin6ales & Valeurs Ve!}orielles Michel
AcpTication ~ la d&rivation
Metivier
Introduction. La th@erie des martingales ~ valeurs dams un espace vectoriel localement convexe g@n@ral est loin d'avoir atteint le m@me @tat d'@laboration que la th@orie des martingales valeurs r@elles. Dans cet expos@, je me propose de donner un apergu de ce qui me parait ~tre l'etat acfuel de la th@orie des martingales vectorielles sulte des travaux de S.D.Chatterji, Hans, J. Neveu et moi-m@me.
(~ base croissante),
telle qu'elle r@-
F.Scalora, A et C. Ionescu Tulcea, M. Drimil et 0.
Je donnerai quelques r@sultats qui n'ont pas encore @t@
publi@s ou qui ont ~t~ ~nonc@s
sans d@monstration (notamment darts ~21] ).
De m@me que darts le cas r@el, la th@orie des martingales peut Stre app~ligu@e au probl@me de la "d@rivation" lativement ~ u n e tribu ~
des fonctions d'ensembles. Nous envisagerons ici la "d@rivation" re I
mesure positive born@e p
de parties d'une ensemble ~
des fonctions d'ensemble d@finies sur une
, ~ valeurs dams un espa~e vectoriel localement
convexe ~. L'existence d'int@grants de Radon-Nikodym,
dams le cas des mesures vectoriel-
les, n'ayant d'ailleurs pas re~u de r@ponse compl@te,
ceci nous permet d'obtenir quelques
r@sultats prolongement ceux c@l@bres de J.Dieudonn@
(cf.[6] et [#]), et pr@cisant d'au-
tres que nous avons pr@c@demment don~@s (cf. [18])
§ 1 - Th@or@mes pour Martingales ~ Valeurs Vect_orielles. I. -
Pr@liminaires - Probl@mes.
1.1 - Propri@t@s scalaires - Propri@t@s faibles. Darts route la suite V d@signera un espace localement convexe en du~lit@ avec ~
(cf. F3]
chap. IV ). Une fonction f d@finie sur un ensemble E, ~ valeurs dans ~', sera dire poss@der scalai-
-
-rement une propri@t& P s i
2 4 0
pour tout v' ~ V'
-
la fonction v' o f
(not@e aussi ~f, v' > )
poss~de la propri6t~ P. Les propri@t6 faibles sont celles qui se d@finissent relativement & la topologie faible
E, E') deX.
(cf. ~3~ chap. ~V)
~.2 - Base de ~rtim~ale. I d@signera toujours un ensemble d'indices ordomm@ par ume relation not6e ~ et filtramt droite pour cette relation. (~,
T , H ) d@signera un espace mesur@ complet, p @tant une mesure positive born@e.
une base de martingale sera toujours pour nous la donn6e, outre (~ ,~- , p famille croissante ( ~ )
) d'ume
~ ~ I de sous tribus de ~-.
~-oo d@signera la compl@tion relativement ~ / ~ d e
la tribu engendr@e par
U ~ ~I
.
1.2 - Martingales faibles ~ valeurs dams V. Nous dirons que le terme ( ~
(MI) Pour t o u t ~
, f~
, f~) ~ z I e s t
ume martingale faible (pour la dualit@
est une fonction d@finie s u r ~
, ~ valeurs dams V, scalairement
-mesurable. (~)
Pour tout ~
et tout F e ~
la forme lin6aire
~ ~F
@16ment de ~ not@
~F
< f~ d p
l'int6grale faible ~F f~ d~est um @lement de ~
f~ ' v' >
d~
s'identifie par la dualit6 ~
(i.e.:
~, ~ ' > ~ un
).
(M3) Pour tout ~ ~j~ et tout F a ~ :
1.3 - Martingales fortes ~ valeurs dams un Banach. Lorsque ~ est un espace de Banach nous dirons que ( ~
, f~ ) ~ ~ I e s t
une martingale
forte si elle v@rifie les conditions (M~), (M~) et (M~) obtemus en remplacamt darts (MI) , (M2) et (I3) la notion de mesurabilit@ scalaire par celle de mesurabilit@ forte (ou mesurabilit@ au sens de [9] par exemple )
et celle d'int@grabilit6 faible par celle
-
d'int6grabilit6
241
-
forte (Int@grabilit6 au sense de S.Bochner
!
cf. ~13 ou [ 93 ).
I1 est 6viden# que toute martingale forte est une martingale faible pour la dualit6 <~,
v' >.
1.~
- Probl@mes.
Soit ( Y~ , f ~ ) @ 6
I une martingale faible (resp.forte).
Probl~me I. Lorsque I n'a pas de plus grand 616merit soit ~ l'ensemble ordonn~ obtenu en ajoutant ~ I un plus grand ~l~ment not~ Existe-t-il une application foo d e ~ d a n s
co, et soit ~-oo comme d~finie en 1.2. V telle que ( Y ~ ,
f~ ) ~
soit une martingale
faible (resp. forte) ? Probl~me II.
Si f ~
eiste comme solution du probl~me I, la famille ( f ~ ) ~ ~ I converge-
t-telle suivant l'ensemble filtrant I pour un mode convenable de convergence vers f ~ ? Probl~me III. IndSpend-mment de l'existence d'une solution du probl~me I, la f~m~lle (f~) ~eI
1 - 4 -
eonverge-t-elle
enun
sens convenable vers une application f de &~ dans V?
Les r6sultats connus darts le cas fort.
Nous r@sumons ici les r@sultats obtenus dans le cas r@el avec leurs analogues (et leurs lacunes) darts le cas des martingales fortes ~ valeurs dams un espace de Banach. a - Cas R@el - (Th@or@mes de Doob et Helms). Le probl@me I admet une solution si et seulement si ( f @ )
est terminalement equi-inte.
grable. La famille ( f = ) ~
e I converge alors (Probl~me II) vers foo
I = N, La convergence a lieu ~galement
dams
L 2 (6~,~-,p)
- Si
p .p.p
Th~oremee de Doob et Krickeber~: Si
su p / J f ~ I d p
~ K <+oo la famille ( f ~ ) ~ G I
f (probl~me III). La convergence a m~me lieu ~ - p . p
si I = N.
converge en mesure vers u n e
fonction
-
2 4 2
-
b - Martingales fortes. ll est bien connu
(voir par example /5] et [25~ qu'une ms~tingale forte
(f~) ~ I
valeurs dans un espace de Bamach B peut tr~s bien ~tre @qui-int@grables sans qu'll existe pour autant de solution au probl@me I. On a toutefois: Th@or@me de Chatter,ji - B c a l o r a . - Tulcea Si B e s t
(Probl@me I)
le dual d'une espace de B~n~ch, et est s@parable, le probl@me I admet une so-
lution pour la martingale forte
(fn' ~ n ) n ~ N
si et seulement si (fn) est @qui-int@-
grable. Le probl@me II fort: est enti@rement r6solu par un th6or@m de A et C Tulcea et Neveu que nous donnons plus loin
(Th. 3)
Enfin pour une martingale forte (fn' @Zn) n a _N ~ valeurs dams un espace de Banach B, le probl@me I I I a
re@u la r@ponse partielle suivante:
Th@or@me de Chatter,~i-Scalcora-Tulcea. Si B e s t
(Probl@me III)
le dual d'un espace de Banach et est s@parable, et si sup/~fnJ dp
<+so, (fn)
converge
~ .p.p vers une fonction f.
Dams la suite nous donnons qulque r6sultats relativement aux martingales faibles et @galement quelques compl@ments sur les martingales fortes.
243
-
-
2. Martin6ale ~ faibles. Nous consid@rons une martingale faible ( ~ ,
f~ ) ~ a I
~ valeurs dams ~ . On designe
ps: ~ un ensemble de parties de E convexes @quilibr@es et compactes pour la relation d(V, V'), filtramt pour la relation d'inclusion c et constituant un recouvrement de E.
Th@or@me 1. On suppose que la martingale ( ~
, f~ ) % g l v@rifie
l'une des conditions (A) ou (B)
suivamte: (A)
I1 existe une suite croissante
une suite croissante Pour tout
~
(~n)
(~n)
extraite de ~-eo et (Qm)
d'indices telles que:
ou (B)
[I?]
et
~ ( U ~ n ) = P (&~) et: n
~ ~n
I~(~) . f ~ ( o ~ )
En outre
extralte de ~
(f~) ~ e Iest
e
Qn
p
.p . p .
(I)
scalairement terminalement uniform6ment int@grable
(cf. [15]
).
I 1 e x i s t e une sous m a r t i n g a l e
r~elle
( ~
, ~)
~ m iterminalement
•
f
unzformement
int~grable et Q E G tels que : Pont tout ~
e I
f~(~)
~
~(~)
Q
~
. p . p .
Alors : I°)
Ii
(f@)%6 2 °)
Si
existe fee telle que (5~% , f ~ ) ~ e 1
~
soit une martingale faible.La famille
converge scalairement en moyenne vers f ~ .
E' admet un ensemble d@nombrable partout dense pour la ~ -topologie, il est
possible d'extraire de I ume suite (~ n ) telle que lim
faible
f ~n (~)
= f&~)
W • P • P •
n - ~ - ()O
3 °)
Si X'
admet um ensemble d@nombrable partout dense pour
la ~ -topologie et
siI=N; lim
faible
fn(~) = f (w)
n-~oo
(I) IF d@signe l'indicateur de l'ensemble F.
H
. p . p .
- 244 -
Pour une d@monstration
de ce th@or~me voir •22] . Ce th§or@me compl@te et pr@cise des
th~or~mes analogues publi§s clans [q8~ et ~2~] des martigales r~ellee r@ele (< f ~ ,
v' >) ~ g I
suite u t i l i s e r u n e Tulcea (cf. ~ 2 ]
< foe
(th@or@me de Helms
de rel@vement,
~oo' P )" ll faut en-
not~mment un th@or@me de A et C Ionescu
) pour montrer l'existence
~.p.p
de la th@orie
~ ~ que pour tout v' a ~' la martingale
converge en moyenne vers u n ~ v , e L ( ~ ,
m@thode
v'> : ~ ,
. I1 r@sulte immediatement
defoo v@rifiant
:
pour tout v'.
Nous donnons @galement u~e r@ponse partielle au probl@me III. Th~or@me 2. I°) (~-~ ' f ~ ) ~ e I
~tant une martingale faible,
l'une des conditions
dessous est suffisante pour que (f~) ~ a I converge scalairement application f de ~ (A') ( ~
dans ~, scalairement
, f~ ) ~ g Iest
(A') ou (B') ci-
en mesure vers une
~oo-mesurable~
scalairement
~ variation born@e ; il existe une suite
(~)
croissante extraite de ~-oo' (Qn) extraite d e ~ n telles que : p ( ~ n ) = ~(~) et
et une suite (O~n) ex~raite de I
n
Pourto~t
~ ~ ~n
lgn ( ~ )
" f~(~)~%
(B') I1 existe une sous-martingale sup / I f g l
dp
= K < + eo et um Q ~ ~-
pour tout ~ g I 2 ) si I ~ N e t on
( ~-~ , % ~ )
f~ (~)
g
~(~)
~" P " P " ~ e I r@elle
, telle que
tel que: . Q
~
. p . p .
si ~' admet un ensemble d@nombrable partout dense pour la ~--topologie
a
lira faible
fn ( ~ )
= f(~)
~ . p . p .
Pour une d@monstration voir [2~ (cf. [16~ )sur la convergence
. La d@monstration utilise le th@or~me de Kricketerg
stochastique
des martingales
r@elles g variation born@e
et des m@thodes de rel~vement analogues g celles utilis&es pour le th~or~me i. 3. - Martingales
fortes.
Dams le n ° 3, ~_ est un espace de Banach, en dualit@ avec V', lorsque no~s utiliserons
- 245 -
le vocable faible pour [,sams pr~ciser [' nous entendrons par la que T' est le dual de l'espace de Bamach ~. Z i par contre [ e s t "faible'" relatif ~ dualit@
~(B',
un dual B' d'espace de Banach B, le mot
B) sera souvent remplac~ par "vague".
3.1 - Passage du cas faible au cas fort . Si ( ~-~ ,f~ ) ~ i
est uae martingale forte, c'est a forteriori une martingale
et on peut dams certains cas lui appliquer les th~or~mes qui precedent. f
faible,
Soit alors
solution du probl~me I faible. Si on peut montrer que foo prend ses valeurs dans
un sous-espace s~parable
de ~, il r~sulte d'un th~or~me de Pettis (cf. [ 2 ~
que fo~ est fortement mesurable.Si
~llfooll d~
<
+
on peut alors prouver que
oo)
on a prouv@ que ( ~ - ~ )
f~ )~
~nonc~s des lemmes permettant
~ est ume martingale forte. Darts [26] , [2o]
et [2~] sont
de faire le passage du cas faible au cas fort.
La solution de probl~me II fort est compl~tem~nt du ~ A e t
et [9] )
C Ionescu T ulcea et J. Neveu
donn~e par un th~or~me tr~s important
:
Th§or~me ~. Si f@o ' solution du probl~me I fort existe,
(f~)
converge vers foo darts L I ( ~ , ~ - t V ) p ) .
Si I = N , on alors ~galement lim forte n --~oo
fn t.oo)~Lwl~ .r~_ P • P •
3.2 - Probl@me I for~. Nous ~noncerons
deux r~sultats.
Pour les d~montrations
voir [ 2 2 ] .
Th@or~me 4. Soit (~-~ , f ~ ) ~ m I ume martingale forte ~ valeurs doDs am espace de Banach s@parable,
de dual [' . Bolt Q une pattie convexe ~quilibr~e
([ ,['). Si l'hypoth~se (;~
de [ compacte pour
(B) du th~or~me 1 est v~rifi~e il existe foo telle que
, f ~ ) ~ ~ ~ soit une martingale forte.
Si [ e s t
um espace r§flexif et si I = N
pour ( A ~ )
la sous-martingale
et F. Scalora.
, en prenant
pour Q la boule unit@ de ~ et
( II f~. lJ ), on obtient les th@or~mes
de S. D. Chatterji
- 246 -
Th~or~me ~. Soit ( ~
, f a ) d G I ume martingale forte g valeurs dams le dual B' d'um espace de
Banach B. Si B' est um espace de Bauach s@parable, suffisante pour qu'il existe foo telle ( ~ est que la famille ( f ~ )
la condition n@cessaire et
, f ~ ) ~ m ~ soit ume martingale forte
g g I soit terminalement uniform~ment int~grable.
Dams le cas I = _N le th@or~me 5 a @t@ ~nonc@ par A et C. Ionescu Tulcea dams I = _Net V r@fl@xif on retrouve le th@or@me de S. D. Chatterji et F. Scalora.
[12] . Si
- 247 -
§ 2 - Application ~ la d~rivation 1.
-
Probl~me de d~rivation globale oour mesures vectorilles.
1.1 - Base de d&rivation. Une base de d&rivation ( ~ , ~. ( ~ 0 ) ~ ~I donn&e d'un espace mesur~ ( ~ , ~ - , p), born@e, et d'une famille ( ~ ) propri~t~s suivantes (~
l)
Iest
(~
2)
~
( ~ 3)
la mesurep @rant toujours ici suppos~e r~elle
~ ~I de partitions finies de~poss~dant
les
: un ensemble ordonn~ par ~ et filtrant ~ droite.
, ~ ~I a v e c ~
~est
) est d~fini dans ce qui suit par la
~
~D~plus fine queJD~.
la completion de la tribu egendr~e par ~ I
JO .
1.2 - Fonctions additives ~ valeurs vectorielles. Nous consid&rons ume fonction ~ d~finie sur a n n e a u ~
~Ji ~D g valeurs darts ~, et
additive s u r ~ . Nous rappelons qu'une fonction~ additive s u r ~ sur ~ la topologie ~(~,~')
est ~ - a d d i t i v e
et ~ -additive lorsque l'on consid~re
pour toute topologie sur ~ compatible avec
la dualit& K ~,~'~ .(cf. [ 9~ chap. IV-10 et ~18~ chap. V. ) Nous pouvons doric dire ~ additif sans pr~ciser. 1.3 - Fonction d' ensembles domin~e par une mesure
. Absolue continuit~ scalaire.
On dira qu'ume fonction d'ensembles ~ (r~elle ou vectorielle) mesure ~ si ~ ( A )
= 0
--~
~(A)
est domin~e p~r la
= O.
On dira que ¢ ~'valeurs vectorielle est scalairement absolument continue par rapport ~
si pour tout v' et tout
il e x i s t e g t e l
que
~(A)
g ~ O
~ ~ ~
~¢(A), v'>I ~ £
I1 est @vident que si ~ est ~ -additive les 2 notions pr@c§dentes coincident. On salt ~galement (voir ~ 6 ~ et ~2@~ ) qu'il n'existe pas de "bonne condition d'absolue continuitY" permettant d'@noncer um th~or~me de Radon Nikodym pour mesures vectorielles,
relativement ~ une mesure r~elle donn~e.
1.~ - D~rivants et d~riv@es d'une fonction 1 d~finie sur A. On pose
-
D ~C¢o) =
Z
Fe~
2 4 8
-
lF ( ~ )
avec la convention ~ j - - ~ =
0 si #(F) =
O
D~est
appel~ le d~rivant de ~ relativement ~ J D .
Si J ~
d~signe la tribu finie engendr~e par~D~ , on voit que si ~ est domin@e
par p , ( D ~ ,
~-~) % e I e s t
ume martingale. C'est mSme une martingale forte si ~ est un
eepace de Bamach. On a d'ailleurs ~ oo =~~i ( D @ ~ ~ ~ zc°nverge pour un mode"T" de convergence vers f, on dit que f es~
la T-d~riv6e de W relativement A la base ( ~ Si ( D ~ ( ~ )
).
~ e I converge faibleme~t (reep. fortement) vers f(o~) on mit que f ( ~ )
est la d@riv@e faible (resp. forte) de ~ au p o i n t ~ . Si f est solution du probl~me I faible (resp. fort) on voit que f e s t
une
deneit@ faible (resp. faible) de ~. Le probl~me I e s t
donc celui de l'existence d'une densit@
Le probl@me III est 6onc celui de l'existence d'une T-d~riv@e Le probl~me II est doric celui de la relation entre densit6 et T-d~rivSe. 2.D6rivation fa%~le. Dane toute cette partie 2 du paragraph 2, nous consid~rerons un espace localement convexe E, en dualit6 avec V__' , et ~ d 6 i g n e r a comme pr&c6dement um ensemble filtrant de parties convexes @quilibr~es de ¥, compactes pour ~(V_,~'). 2. l- Existence d'une densit6 faible et d6riv@s correspondants. Th6or~me 6. Soit ~ une fonctionadditive d'ensembles definie sur ~ , ~ valeurs dams V, domin6e par pet
poss~damt l'une des propi~t~s (A'') ou (C'') suivantes: (A'') ~ es#
~I
~
~ -additive et il existe use suite ( ~ n )
st une suite ( ~ ) extraite de 6~telles que
F ~ ~
W(FQ~n
) e
y(zn~n)
p(
~n
croissamte extraite de ) = p(~)
et :
. Qn
(B'') Ii exlste use mesure r@lle born@e Q , absolument continue par rapport ~ ~ ,
- 249 -
, et um Q e ~
tels que : F
~
~-
~(F)
~
~(F)
. Q
Alors z I ° ) - ~admet
ume densit@ faible f relativement
2 °) - les d&riv@es
(D ~
) ~ ~ I
(D ~
lim n
faible
D~
--~ 0 o
en moyenne vers f.
d@nombrable partout dense pour la ~-topologie,
) n
.
convergent suivant I scalairement
3 °) - si _V' admet um sous-ensemble existe u~e suite
~p
extraite de la f~mille des d@rivants,
il
telle que
n~N
(~)
= f(o~)
p
. p . p .
n
4 °) - si I e s t
d@nombrable ~ admet pour d@riv@e faible f, en # -presque tout ~ 6 % Z ,
relativement
~ la base
(~n)
de d~rivation.
D@monstration: Remarquons
que sm ~ est une mesure r@elle born@e absolument continue par rapport ~ ~ ,
la martingale
(D~ , ~-~ )
r@elle est uniform@ment
int@grable
(et converge d'ailleurs
en moyenne vers la densit@ de ~ relativement ~ p ). Si l'hypoth~se
D#
Q
.p.p
et le th@or~me r@sulte imm@diatement (B)
QB'') est vraie,on a:
de l'application
duth&or~me
1 lorsque l'hypoth~se
est vraie .
Lcrsque (< D ~
(A' ')
est vraie, ~ @ t a n t ~-additive,
, v' > , ~
r@elle b o r n ~ e < ~ ,
) ~al
r@elle
est constitu@e par la famille des d@rivants de la mesure
v'> . On sait (cf.
est de variation born@e et absolument terminalement uniform@ment
pour tout v'a V' la martingale
int@grable.
~$7S P. #85 prop. 2.1.2.)
qu'une telle martingale est
continue doric (ibid. prop. 2.3.2) qu'elle est Le th@or@me r@sulte encore darts ce cas imm@di-
atement du th@or@me 1. Cas particulier. I°)
Si
B''
est un dual
les boules de n@ralisant 2°)
B'
d'espace de Banach
B', qui sont compactes pour
celui de J. Dieudonn@
B. En prenant pour @l~ments de
e-
o-(B' ,B), on obtient un th@or~m~ g@-
(cf.~6J) ou B est suppose s@parable.
Si B'' est um dual _~ d'espace de Banach B, l'hypothese (B'') est vraie si ~ est
- 250 -
variation oorn~e (au sens de [9] chap. III), car on a ~ F )
~ var W(F)
. Q si Q d~signe
la boule unit& de B'. 2.2 - Un th~or~me de d @ c o m ~ t ~ n
c__~o_mm_e_~lication du th@or@me 2.
Th@or~me 7. Soit~une
fonction additive d'ensembles d~finie s u r ~ ,
par H et poss~dant
~ valeurs dams T , domin~e
l'une des propri~t~s (A''') ou (B''') suivantes:
(A''') ~ est scalairement ~ variation born~e et il existe une suite ( ~ n ) croissamte extraite de H(U~n)
=
H(~)
W ~ a ~-
~LJei ~
et une suite ( ~ )
extraite de
~ telles que
et I
~ ( P n @..n) ~ P ( F o ~ . n )
. Qn
(B''') I1 existe une fonction simplement additive, ~ variation born~e ~ sur ~ , domin@e par p
, et um Q a E
F ~ 3r ~(F)~
tel que :
~ (F). Q
Alors : "I ) ~ est d§composable de fa~on unique en la somme de deux fonctions additives 1 et ~ 2 poss@dant les propri&t&s suivantes: -~l
admet ume densit§ fl par rapport ~ /J
-72
est scalairement purement simplement additive (i.e. ~ v ' a
soit la mesure ~ -additive
H'
: I p'l
V' quelle que
_L]<¢2, v' >I~' ~ ' = O).
2 ) Les d§riv~es ( D ~ ) ~ ~ i convergent suivant I scalairement en mesure vers f 3 ) Si V' admet un ensemble d~nombrable partout dense pour l a g si I = N , ~ admet fl(oO ) pour d~riv~e en ~ -presque tout point ~ E ~ ,
-topologie et relativement
la base de d~rivation ( ~ n ) . Demonstration. Si ~ admet une d~composition du type indiqu~,~
v'~V'<~,
v'>
=<~l'
v'> + < ~ 2 '
v'>
est la d~composition unique de la fonction r~elle simplement additive ~ variation born~e *~, v'>
en sa partie 8--additive (ici absolument continue par rapport ~ H )
et en sa partie purement simplement additive. Si ume autre d@composition
-
25~
-
= ~'I + ~'2 existait, on aurait donc w ~'2' v' > = ~W2 ' v' > pour tout v' d'od ~ 2 = ~'2" D'od l'unicit~. En appliquant le th~or~me 2 ~ la martingale ( D ~ )
~ai
on voit que cette martingale
converge scalairement en mesure vers une fonction fl" Pour tout v'~ ~' la martingale r~elle(< D ~ ,
v' ~ ) ~ a I converge stochastiquement vers la densit~ de la partie
absolument continue de <~, v'~ (< fl' v'> est donc cette densitY) On sait que F
~
~
(<~(F),
, v'> .
v'>
-
%K
fl' v'~ d ~
est la pattie purement simplement additive de
(cf. [~7S th. 3 P. ~9q ). On voit donc qu'en posant ~l(A) = ~(A) - ~Afl d~
on obtient la d~composition voulue, ainsi d'ailleurs que la deuxi~me partie du th~or~me. La 3~me partie r~sulte alors imm~diatement du th~or~me 2. $ ). 3. D@rivation forte. 3.1 - Un cas d'e.~i,stence d'ume densit6 forte et d@riv~ correspondants. Th~or4me 8. Soit~
~-additive
~ valeurs dans un dual s@parable B__' d'espace de Banach B. On suppose
que ~ est ~ variation forte born&e par la mesure~. Alors il existe une densit~ forte f de ~ relativement ~ la m e s u r e p .
La f~mmille des
d@rivants ( D ~ ) ~ ~ I converge vers f darts Ll(g~,~-,p ,B'). Si I e s t
d~nombrable
~ admet f ( ~ )
pour d~riv@e forte en p ~presque tout c o ~ Z .
D6monstration. L'hypoth~se de variation forte born~e s'~crit en effet: + ~
~ s~p
F~- H
~ (V)I# = s ~ p f
La variation~ = Var~ de I e s t
I# D ~
#I
dH
une mesure r~elle born~e domin~e p a r ~ ,
absolument continue par rapport ~
. La suite des d6rivants ( D J )
~ = Var~ est
est donc une
martingale unifo~m~ment int~grable ( cf. ~qSS ) qui majore terme (lJ D ~ U Le th~or~me 8 est donc consequence
).
du th~or~me 5.
3.2 - Un th~or~me de d6composition. Soit S u n ensemble portant la pattie absolument contenue de Var~ par rapport ~ ~ et tel que SC~orte la partie de Var~ ~trang~re ~ ~ .
- 252 -
On apellera F - ~ ~ (So F) la partie absolument continue de 5o relativement ~ P et F --~ ~SC~
F) la pattie ~trang~re de ~ .
On a la proposition ~vidente. Proposition. Une mesure ~ ~ variation forte born@e est domin@e par p s i de ~ 6trang~re ~ p e s t
et seulement si la partie
nulle.
Th~or@me ~. Soit ~ R valeurs dams un dual s@parable B' d'espace de Banach B. On suppose que ~es~ de variation forte Var~ born@e. On suppose @galement que I e s t
d@nombrable.
Alors la suite des d@rivants ( D ~ )
converge ~-presque partout pour la topologie forte
dams B' vers une densit~ forte de la partie absolument continue de ~ relativement ~ ~ . Demonstration. On a en effet
=
~
Iv
~(F)
Vary(F)
Posons
Va~ ~ (~) v ~ (gn ' ~ n ) est une martingale vectorielle forte ~ valeurs dams la boule unit@ de BI'. Elle converge fortement p _presque partout vers une densit@ forte g de ~ par apport Var~ . La sous-martingale r@elle A n , S - n )
converge ~ - p r e s q u e partout vers une densit@
de la partie absolument continue p
de Var~
relativement ~ H
.
La suite (Dn~ des d@rivants converge donc ~ -presque partout vers ~ .g pour la topologie forte de B'. Pour tout F¢ ~ on a en dSsignant par S u n
ensemble portant la partie abeolument continue
de ~ tandis que S c porte la partie &trang@re & p
.
-
253-
Z)'o~ le th6or~me. Nous terminons
en donnant une cons6quence
du th6or~me 4 m
Th@or~me 10. Bolt ~ A valeurs dans un espace de B~uach B. On suppose ou bien que I e s t ou bien que B e s t
d~embrable,
s6parable.
Si il existe une pattie faiblement compact Qde B telle que
alere i ~ admet une densit@ forte f rapport ~ p
, la famille des d~rivants ( ~ ~ ) ~ ~ I
converge vers f dana L1(a,Y~p, ~), e~ si I = N~admet f ( ~ ) pour d@riv@e forte en p presque tout
(u ~ 2~
.
- 254 -
BI (1)
BOCHNER S.
BLI
0 GRAPHI
Integration von Flmktionen,
E
deren Werte die Elemente einer Vektor-
raGmer sind. Fundamenta Mathematicae. Vol 20 (1933) p. 262-276. (2)
BOCHNER S.
Partial Ordering in the theory of Martingales Annals of Math. Vol. 62-n°1. July 1955 P. 162- 169.
(3)
BOURBAKI
Espaces vectoriels topologiques.
Paris Hermann (1955).
(4)
BOURBAKI
Integration. Chap. VI Paris Hermann (1959).
(5)
CHATTERJI S. D. Martingales of Banach-valued random variables. Bull. Amer. Math. Soc. 66 (1960), 395-398.
(6)
DIEUDONNE J.
Sur le th@or@me de Lebesgue Nikodym.
(V) Can. J. of Math. vol.
III N ° 2 1951 p. 129-139. (7)
DRIHL M. and HANS 0.
Codtional
expeotations
for generalized ramdom variables.
Trans. Second.Prague Conf. ou Info. Theory. Prague 1960. (8)
D00B J . I .
Stochastic processes. New York 1950.
(9)
DUNFORD and SCHWARTZ
Linear operators Part. I - New York 1958.
(10)
MOURIER E. El@ments al@atoires dans un espace de Banach.
(11)
HET,M~ L . L .
Mean convergence of martingales.
Trans. Am. Math. Soc. Vol. 87
1958 p. 439-4@5. (12)
IONESCU TULCEA A. et C.
On the lifting property I. J. of Math. An. and Appl. 3 1961 p. 537-546
(13)
IONESCU TULCEA A. et C.
On the lifting Property II. J. Math. Mech. 1962 773-795
(1@~
IONESCU TULCEA A. et C.
Abstract
ergodic theorems. Trans. Amer. Math. Soc. 107
1963 p~ 107-i25. (15)
KRICKEBERG K.
Convergence of Martingales with a directed index Trans. of the Amer. Math. Soc. Vol. 83 n
(16)
KRICKEBERG K.
2 p. 313-317.
Stochastische Konvergenz yon Semimartingalen.
Math •eitschr. 66
1957 P. 470-486 (17)
KRICKEBERG K. et PAUC CHR.
Martingales et d@rivation.
Bull. Soc. de France 1963
p. 455-543 (18)
METIVIER
Limites projectives de mesures. Martingales. Applications Annali di
- 255 -
Mat. Pura ed Appl. (IV) Vol. LYT 1963 p. 225-352.
(19) (20)
METIVIER M. METIVIER M.
~artingales ~ Valeurs Vectorilles.
Bulletin Soc. Math. Gr@ce 1964.
Martingale nit Werten in einem lokal-konvexen Raum: exposes dactylographi@s au S@minaire de Probabilit@s
de l'Universit~ de Hambourg
(Mai - Juin 1965.) (21)
METIVIER M.
Martingales ~ valeurs dams un espace localement convexe. A
paraitre.
(22)
~KETI~IER H.
Martingales faibles et Martingales fortes. C. R. Acad. Sc. Paris t. 261 p. 3 723- 3 726.
(23)
NEVEU I.
•elation entre la th@orie des martingales et la th@orie ergodique. Colloque International de Th6orie du Potentiel Paris
(24)
Juin 1964
PETTIS B. J .0n integration in vector spaces. Trams. Amer. Math. Soc 4@ "1938 p. 277-304
(25)
RONNOV U.
Martingales ~ valeurs vectorielles et d@rivation. S@m. de Th@orie des Probabilit@s.
(26)
SCALORA ¥.
Instinct Henri Poincar@ ~nn@e 64-b5.
Abstract martingale convergence theorems. Pac. J. of Math. Vol. II 1961 n ° 1 p. 347-374.
- 256 -
ATOMES CONDITIONNELS D'ESPACES DE PROBALITE ET THEORIE DE L'IN~O~JATION J. NEVEU
I At omes conditonnels d'un espace de prebabilit@. Soit
(~, ~, P )
not@e~.
u n espace de probabilit~ dont la classe des ensembles n@gligeables
sera
Dans route la suite deux ensembles de ~ en deux fonction r @ e l l e s ~ - m e s u r a b l e s
ne diff@rant que sur un ensemble n@gligeable seront toujours identifi@s d'@galit@,
; les relations
d'inclusion ou d'in@galit@ devront donc toujours @tre comprises comme des
relations m o d u l o ~ .
Lorsqu'on ne consid@re comme nous le raisons ici les ensembles de
que par leur classe d'@quivalence, on salt que route famille d'ensembles dans O , d@nombrable ou non,
poss~ de
bornes sup@rieure et inf@rieure que l'on qualifie de
bornes essentielles. Enfin nous ne distinguerons pas non plus dans la suite deux s o u s - ~ -algSres de ~ @gales modulo o~ ~. Darts le travail E2~
@crit en collaboration avec A. Hanen, nous avons g~n~ralis~ les no-
tions classiques d'atome
et de partie atomique d'um espace de probalit@ de la mani@re
suivante. D@finition 1 : Etant donn@ une sous ~ - a l g ~ b r e ~ d e appel@e atome c o n d i t i o ~ e l An ~ (~ ~ ~
de~
sont identiques m o d u l o ~ . P )
~ , une partie A d e ~ A d a n s
par rapport ~ ~ l o r s q u e La pa~tie ~ - a t o m i q u e
~ est
les g -al~@bres traces A o ~ et de l'espace de probabilit@
est alors d~finie comme la borne suo@rieure essentielle des atomes con-
ditionnels de ~ p a r rapport ~ B . Lorsque ~ se r@duit ~ la ~ classiques
-alg@bre ~ = { ~ , ~ I '
on retrouve effectivement les notions
d'atomes et de partie atomique de l'espace de probabilit@
(~,~,
P ) . Le
seul r@sultat essentiel de l'@tude des atomes d'un espace de probabilit@ qui ne se g@n~ralis@ pas aux atomes conditionnels est sans doute le suivant distincts sont n&cessairement ditionnels distincts
disjoints, il n'en est pas de m~me de deux atomes con-
(n@anmoins comme tout sous-ensemble d a n s ~ d ' u n
par rapport ~ e s t
: alors que 2 atomes
atome conditionnel de
encore un tel atome, l'intersection de deux atomes conditionnels
- 257 -
est encore un atome conditionnel
).
L'@tude des atomes conditionnels d'un espace de probabilit@
nous parait se faire le plus
simplement ~ partir des deux lemmes suivants. Dams la suite nous d@signerons par E l'esperance
conditionnele
g@ngralement P ~ (A) Lemme 1 ~ ~
par rapport g la sous- ~ - a l g @ b r e ~
au lieu de
D@monstration
dans~qui
contienne p. s. un ensemble
P g ( A ) ~ o~ . Toute propri@t@ ~ - m e s u r a b l e
l'est donc aussi s u r { P ~ ( A ) >
nous @crirons
E ~(1A).
plus petit ensemble ( m o d u l o ~
donn@ A de ~ est @~al g~
de ~ e t
valable p. s. sur A
0~.
:
Pour tout B g d8 , on a plus grand ensemble
P(AB) =
P~(A)
(modulo~g~
dP ;il en r@sulte que ~P ~ ( A )
= O} est le
dans~8 qui soit disjoint de A . Le lemme s'obtient par
passage aux compl@mentaires. Lemme 2 : Pour tout atome conditionnel positive f-mesurable
par rapport ~ ~ ,
EJ~( f 1A ) = f PJS(A) I1 s'en suit que si
A
_et
A'
A
de ~ par rapport ~ ~ et toute fonction r@elle
on a : sur
A .
sont deux atomes conditionnels
de par rapport ~ ~8 m on
a : P~(A)
D@monstration
= P~(A')
sur
A A'
:
Comme les fonction r@elles d~finies sur exac~ement
les restrictions
rapport ~ J8 , l'hypoth~se
A
et mesurables par rapport ~
AM
~
= A ~ J8 entraine l'existence f = g p.s.
E~8(f 1 A) = E ~ ( g
et sur
1 A) = g P ~ ( A )
A
sur
ensemble
A'
P~(A)
dams ~ ; si
sur A'
A
donc que
d'une fonction
p~(A
g
me-
A . On a alors
le dernier membre est @gal
La premiere pattie du lemme est ainsi d@montr@e. A') = 1A,
sont
~ A des fonctions r~elles d@finies sur ~CA et mesurables par
surable par rapport ~ d8 telle que
P~(A
AO~
z PB(A)
On en d@duit que A') = P ~ ( A )
sur
est aussi un atome conditionnel,
A A'
on a
,pour tout
.
-- 258 -
P~(A
A') = p ~ ( A ' )
sur
par un raisonnement sym@trique et la deuxi~me partie en
A A'
r@sulte. Le th§or@me suivant donne une description que nous croyons compl@te de la partie -atomique de l'espace de probabilit@ par rapport ~ d e
(o%, ~, P )
cet espace de probabilit@.
ont @t@ @nonc@es dans[
et des atomes conditionnels
Les deux premi@re parties de ce th&or@me
].
Th~or@me I. I)
La pattie ~ -atomique de l'espace de probabilit@
( ~ , ~ , P ), soit 2 o ,
peut
s'6crire comme la somme d'une f~mille d@nombrable d'atomes conditionnels par rapport ~
deux ~ deux disjoints, soit ~ o = ~i
g§n&ral). Tout sous-ensemble pr~c@dente de ~ o
A
de ~Z o
f
dams ~ admet en fonction de la d@composition
la repr@sentation essentiellement unique
sont des sous-ensembles ~ - m e s u r a b l e s r@elle
A i (cette r@pr@sentation n'est pas unique en
~-mesurable
de { P ~ (A i) > o ~
A = ~
2)
o~
3)
A
de ~ soit un atome conditionnelpar
(i a I)
A =
~I A iBi
rapport & ~
, il faut les
de l'alin~a pr§c@dent,
soient disjoints deux ~ deux.
I1 est toujours possible d'@crire ~ggo
IAn~ , n ~ l j ?
et nulles en dehors
.
et il suffit que dams la repr@sentation Bi
Bi
peut s'@crire surO~ o sous la forme essentiellement unique
Pour qu'un ensemble
ensembles
od les
; semblablement route fonction
f = ~I "IAi gio~ les gi sont des fonctions r@elles ~-mesurables de{ P ~ ( A i ) ~
Ai Bi
comme la somme dZ ° = n~ -~1
d' atomes eonditionnels tels que
A_~ P ~ ( A q~) ~ P ~ (_~.~
A~ m d'une suite
. .. sur ~
.
D@monstration: 1)
La borne sup@rieure essentielle d'une famille arbitraire d'ensembles est p. s.
@gale ~ la r@union d'une sous-fam~lle d@nombrable de la famille donn&e ~ la pattie -atomique d~ o { An ,
peut doric s'@crire comme la r@union d'une suite d@nombrable
n ~ 11 d'atomes conditionnels.
Comme tout sous-ensemble dans ~ d'un atome con-
ditionnel est joints 2 ~ 2 en les remplasan~ ~ventuellement par les Si
f
est une fonction r@elle
O[ -mesurable,
soient
gi
(ie I)
An -
m~n
les fonctions
Am •
- 259 -
-mesurables res~ectivement gi = E ~ ( f
1Ai)/P~(A i)
D'apr~s le lemme 2, on a gi outre si f = 1A P~(Ai)>'0 }
d@finies par : sur
= f
{ P ~ ( A i)> O)
sur
Ai
la fonction~-mesurable ; comme
gi = 0
d'un ensemble ~-mesurable
sur
, = 0
ailleurs.
et par suite f = ~iIAi gi vaut
{ P~(Ai)
0
ou
1
sur
f =
~
2)
Si
A
1A i gi s u r ~ o
Mais de
gi = g~
sur
P~(Aj
la ~ e ~ @ m e
A i B i ; par suite Bj) = 0
P ~ ( A i B i) = 1Bi
{ P B ( A i B i)> O] = B i
A i ,donc
P~(Aj
P ~ ( A i)
et de
Soit {An, n > l }
est une
i)
=
, A contenu
dans~Z o et AIB i
pattie du lemme 2 montre que
P ~ (A) =
Bj) = 0
sur AiB i
si
Bic{ P S ( A i) > 0 }
que si les ~i Ai B i
B i (i a I)
i ~ jet
et de somme convergente,
I~1 = ml ..... ~k = ink} n
et
on en d~-
i ~ j on tire que
B i Bj = ~
si
i ~ j .
sont des ensembles ~S-
est un atome conditionnel.
une suite d@nombrable d'atomes conditionnels 2 ~ 2 disjoints
-mesurables ~ valeurs enti@res positives,
od
0 } et
o}.
de somme @gale g pattie ~8-atomique~-A o" Comme les fonctions ~-mesurables sont non-n@gatives
1Bi
A
et ce qui pr@c@de montre donc que
-mesurables 2 ~ 2 disjoints, l'ensemble
g~
sur~ P ~ ( A i ) >
sur ~ P ~ ( A i Bi)>0}pour tout couple
Inversement on v@rifie imm~diatement
3)
sur
1Ai
A = ~I A i B i d'apr@s ce qui pr@c@de. Comme
sont des atom,s conditionnels,
duit que
on a
est un atome conditionnel de ~ par rapport ~
P~(A i B i )
, doric sur
il reste A montrer que
est unique. Or si ~i
sur { P gi : 0 = gi' par hypothese "
admet la repr@sentation
o . En
contsnu dans{ P ~ ( A i) > 0 ) .
deuxi@me telle repr@sentation, par suite sur~Apuisque
~
sur~
= 0 ~ , la fonction indicatrice
Pour achever la d@monstration de la premiere pattie du th@or@me; la repr@sentation
gi
on air
il est possible de d@finir des fonctions soient ~'i' H 2 '
P~(Aml)~P~(Am2)~
est un entier arbitraire distinct de
P~(Am)
"'"
telles que sur
... ~P~(Amk)>~ P~S(A n)
m 1 ...... m k . Alors A k : ~m { p k : m
} Am
- 260 -
est um atome conditionnel en vertu de l'alin~a (2) et lee 2 . En outre
{P~(~)
=
~
~
m
sont disjoints 2
P ~ (Am) ce qui montre que
IF k = ml
le k~me maximum de la suite I P ~(Am)
~ ( k ~1)
, m ~I]
P~ (~)
et par suite que la suite {
est p.s. non eroissante sur 6~o; on voit en outre que ~-k P ~(Ak~ = ~ m ce qui suffit ~ entrainer que ~ k
est
P ~ (~),
k~'l}
p~5(Am) = p S ( ~ )
A~ = 6~o .
Mentionnone enfin le th~or~me suivamt d~montr~ dams [23 . Th~or@me 2 : Quel que soit l'ensemble f
telle que
conditonuel
0-
et par suite
dams G e t
la fonction r~elle ~5-mesurable
, il existe un sous-ensemble
A
de
C
dams ~ et un atone
de ~ par rapport ~ ~5 contenu dams (C - A) tels que
P~(A)~ En particulier si
C
C
P~(A)
f~P~(A
+ A')
est disjoint de la pattie ~-atomique
de
(6~,~, P )
on
A' =
= f .
Fomctions d'incertitude. D~finition 2 : Etant donn~ une sous ~-alg~bre~de fonction d'incertitude d e . p a r
J~(~)
=
En vertu du lense 2 on a
A
{ ~ ~ (~)~
~" }
de ~ par rapport ~
de la partie ~-atomique P~(A)
conditionnels et la fonction P~ (A)>0
soit ~ B
(~),
en posant :
, et en posant
+
sur le compl~mentaire ~ c
outre on a
rapport ~ ,
(~I, 5, P) on d~finit la
= log [1/P~(A>]
sur tout atome conditionnel
~(~)
l'espace de probabilit@
sur
= P~(A')
~ ~(~) A
6~ o
de
(6~, ~ , P ).
sur l'intersection
A A'
de deux atones
est donc dSfinie sans ambiguit~ sur ~ o" En
d'apr~s le lense 1 et on en d~duit que l'ensemble
set pr6cis~ment ~gal ~ la partie J~-atomique de
On peut donner ume d~finition intrins~que de la fonctiom ~ ~ ( ~ )
(6~, ~ , P ). comme suit :
-
Lemme 3 : La fonction d'incertitude
261
~(~) -
-
est la plus petite fonction ~ -mesurable
valeurs dams [0, ~oU telle que l'on ait :
pour route fonction r@elle positive ~-mesurable f. D~monstration : En utilisant le lemme 2 on volt que l'on a sur tout atome conditionnel ES(f)~E~(fl E a outre pour
A) = f P R ( A )
exp ~ - ~
A :
m (~)U
f = 1A, le premier et le dernier membre sont @gaux sur
A . On en d@-
duit que l'in@galit~ du lemme est v@rifi~ sur la partie ~-atomique 6~ o et que cette in~galit@ es~ s u r ~ o
de
(~L,~,P)
la meilleure possible•
L'in@galit~ du lemme est clairement v@rifi@ s u r ~ oc
puisque exp ~ -~ 2
(~)]
= 0
sur~Q 0c • D'autre part comme le th@or@me 2 montre que pour tout partition { ~ ..... Am~
de ~ o c
da~a
telle que
n ~ 1 il existe u~e ~ ~ ( ~ ) , une in~galit~ P ~ ( A k) = ~ P
E ~ (f) ~ f e x p ~ - ~ ] du type de celle du lemme me oeut ~tre v@rifi@ pour route fonction f
positive que si a = + ~ sur ~ 0
~(~)~
~
Ak
~o
; il s'en suit que
D@finition ~ : Etamt donn@ deux ~ - a l g @ b r e s ~ , ~' telles que ~ c ~ ' , ~comme
de ~ d a m s
l'espace de probabilit@
on d@finit la fonction d'incertitude ~ ( d S ' )
la fonction d'imcertitude de l'espace de probabilit@
strlction ~
~'
de
(~,~,
(~,~,
de ~B'
par rapport
(6h, ~''P~3 ') re-
P ) par rapport ~ .
Les fonctions d'incertitude jouissent de la propri@t~ fondamentale d'additivit8 suivante. Th6or@me ~ : Si ~ c ~ ' c ~ "
D6monstration
sont trois sous ~-alg~bres de ~t dams
P )
(6~, ~, P ) on a :
- 262 -
a)
Si
A
est um atome conditionnel
conditionnel
de ~''
par rapport ~
de 55'
~',
par rapport ~ ~ e t
alors
A A'
si
A'
est un atome
est un atome conditionnel
de
~
''
par rapport ~ ~ puisque l'on a modulo u4P: A A' ~
~''
= A(A' ~
55'') = A ( A ' ~
5~') = A'(A ~ 95') = A ' ( A o ~ )
en outre le lemme 2 appliqu@ K l'atome conditionnel P~(A sur A , d o n c
A') = E ~ / P S ' ( A ' ) .
en particulair sur
(resp. la pattie
(resp. 61 o' )
~'-atomique
tenu dams la partie ~ - a t o m i q u e
=
b)
de
SCS,)
= A n ~5 ' = A ~ ~ ''
implique que
. PS(A)
d@signe la partie 2 - a t o m i q u e
de ( ~ , 2 ' ' ) ,
(~% ,55' ')
l'ensemble / A o 6 l o'
modulo~.
o
A
de $5'
il r~sulte de l'@galit@ tel que ~'
.
I1 en r~sulte d'abord que
A
est un atome condi-
{P~'(A)>0}
est un atome
par rapport ~J5 (ce qui n'est pas le cas en g~n@ral de
n'appartient pas ~ J8'); si
A'
est un sous-ensemble
A ~ ~8' = A o ~5 [o4P3
A A' = A B. On alors
est con-
de 18 '' par rapport ~ $3 , on a n@cessairement
tionnel de ~5'' par rapport ~/5'. ~ontrons ensuite que conditionnel
de
et que
+
8i A est un atome conditionnel
An~
= P~'(A')
.
A A'.
Ce qui pr@c@de montre que si ~ o (6%,~')
qA ]
A
= A A'~5
de { P ~ ' ( A ) >
qu'il existe un ensemble
1A, P ~' = 1BP ~' (A)
puisque
A'
et
B
A
puisque
O}
dams
/5',
B
dams
~5
appartiennent
et par cons@quent A' { P ~' (A) > O} = B { P ~ 3 ' ( A ) > O } P ~' (A)> O } q
$5, = IP ~' ( A ) > O ]
, ce qui d@montre bien que O ~
•
Le r~sultat de cet alin@a entraine que la pattie ~5-atomique tenue dams 6~ ° 6~ o'
(61, ~5'')
est con-
et donc, en vertu de la premi@re partie de la d~monstration,
identique ~ 61 0 ~l~ , Ainsi ~ 5 ( ~ 5 ' ' )
Corollaire
de
= + co = ~ ( ~ ' )
+ ~'(JB'')
en dehors de
:
La fonction d'incertitude I
( ~' )
d@finie lorsque /5 c J~'
est
est une fonctionelle
- 263-
croissante de ~ '
et d~croissamte d e ~
.
Th@or~me ~.: S o i t ~ u n e sous ~-alg~bre de,dams
(~,~,
P )
et s o i t { ~ ' ~
un ensemble filtrant
croissant, resp. d~croissamt, de sous ~ -alb~bre de ~ c o n t e n a n t ~
. On a alors dams
le cas croissant : lim~ ~ si~'
( ~k
) =
~(65')
sur
~'~
d~signe la ~ -alg~re engendr~e par les~'~ , tandis que dams le cas d~croissant : lim~ ~ ~
si ~ ' '
( ~
' ) =
0~"~(~ ' ' )
d~signe la ~-alg~bre intersection des
U { ~''~ ( ~ )
sur ~
< co }
.
D~monstration : a)
Le corollaire au th~or~me precedent entraine que dams le cas d'une famille ~ "
creissamte que ~ ~ ~5 ( ~ , ) } jor~e par
~ ~ (~')
set une famille filtrante croissants de fonctions ma-
•
D'apr~s le lemme, on a pour route fonction r~ellle positive 38' d~ un
~mesurable pour o
~o fix~ :
E~(f)>f
exp [ - ~ ; ( 3 B
'
ct o
)]~
f
exp [ - l i m ~ o~.
~ (~'~)]
•
L'in~galit~ entre lee membres extremes est ind~pendamte du OtofiX~ ; on voit doric facilement qu'elle reste valable pour toute fonction r~elle positive ~'-mesurable. Mais alors le lemme 3 entraine que ~ (
~ ' ) ~ lim ~ ~
(~'~)
; or d'apr~s le d~but, cette in-
~galit~ ne peut Stre qu'une ~galit~. c)
Consid@rons ensuite le cas d'une famille d~croissante ~ ' I
conditionnel de ~8' ~ o l'ensemble ~ P ~
par rapport ~ 38 pour un °6o
(A)>0}
. Soit
fix@. Pour tout
est un arose conditionnel d e ~ '
A
~ > oco
par rapport ~
un atome , d'apr~s
la deuxi~me pattie de la d~monstration du th§or~me. Comme d'autre part d'apr~s le lemme I, l'ensemble tient
A
~P ~
(A) > 0 3
set le plus petit ensemble dams ~ '
on voit imm~diatement que ces ensemble forment lorsque ~ .~ ~ o
qui conune famille
filtrante croissante d'ensembles dont la limite est n&cessairement le plus petit ensemble dams ~8''
qui contient
A
, c'est ~ dire ~ P ~ ' '
(A)~O} .
I1 r~sulte de ce qui precede que sur exp[- ~ ~ (~)]
= P~
~Ip ~"
A
264 -
on a, iorsque
~>" ~o
:
(A)-~U~] ZP ~F/P~"(A) > 0,~] = ex'p[-~"&(~'')]
On en d~duit que
~o Corollaire Si~
:
' est une sous ~ -alg~bre de (~aans l'espace de pro~abilit&
~ de ~3'
}
(~,~,
est une famille filtrante croissante (resp. d~croissante) on a respectivement lim~
]~
dans le cas croissant (~')=
et dams le cas d6croissant
~(dB')
P )
de s o u s ~
et si
-alg~bres
:
sur
: .
la ~ - a l g ~ b r e
~ ~tant la limite des ~
D~monstration
:
ur
.
Les formules pr&c~dentes sent des consequences imm@diates des t h @ o ~ m e s
3 et
~ .
Etendons maintenant le domaine de d~finition des fonctions d'incertitude de ~ des couples quelcoques ~ , ~'
de sous ~ -alg~bres d e ~
(~')
En dehors du cas ~ c ~ 3 '
auquel on s'~tait limit~ jusqu' ~ present, on consid~rera surtout le c a s o d
~8'
est
engendr~e par une partition d§nomorable d e ~ h d a n s ~ . D~finition ~ : Etant donn@ deux sous ~ -alg~bres arbitraires 4 ,
~
de ~ d a n s
finit la fonction d'incertitude de ~ par rapport ~ s o i t
J~)
J~
(~A,~, Q~)
P )
, on d~-
en posant
= )~(~V~).
Lorsque la ~-alg~bre
~
engendr~e par une partition d~nombrable d e ~ h d a n s ~ q u e
d~signerons alors par q z ( ~ ) , conditionnels de ~ S V ~
les ensem01es de la partition ~ ( 7 )
sent des atomes
par rapport ~ ~ comme on le v~rifie imm~diatement;
suit que l'on a alors sur S~ :
nous
il s'en
- 265 -
Ii n'y a auoune difficult@ ~ ~tendre la propri@t@ d'additivit@ des fonctions d'incertirude : s i 2
et ~ c ~
et la d@finition
sont trois sous ~-alg@bres
ci-dessus
de ~ d a n s
(~,~,
entrainent que l'on a : # ~ ( ~ )
P ), le th@or@me
= J~(S)
+
J ~v~
(~).
On pourra ~tendre de meme le domaine de validit@ du th@or@me # . Par contre le r@sultat suivant exige une d@monstatration nouvelle.(cf.[~ Th@or@me ~ : Soit { ~ n '
n~l]
une suite croissante de s o u s ~
soit~ une sous ~ - a l g @ b r e E[~I
(~,d~, P )
et
de ~ engendr@e par une partition d@nombrable telle que
( ~ ) ~ < oo. Alors la suite
{ ~ ~n(~),
au smms p.s. et au sens de la convergence de engendr@e par lee
-alg@bres de ~ d a n s
~Bn(n~1)
; en outre
n~'l]
converge vers ~ ~ c o ( ~ )
L1$n , si JSoD
E[ sup~
(~)~
~
d@signe la ~ - a l g ~ b r e + oo.
n
D@monstration
s
Pour tout ensemble lim
F
de la partition W ( ~ ) ,
P ~m(F) = P ~
(F)
sur~
le th@or~me des martingales montre que
au sens p.s. ; on en d~duit imm@diatement que
n
lim n
~n(~)
=
~oo(~
)
p.s. sur chaque F a
~q ( ~ )
et par suite s u r ~
. I1 reste
J
alors ~ montrer que
sup ~ ~ n ( ~ )
la covergence dams
L1
est int@grable d@s que ~ ~1( ~ )
des fonctions positives
~n(~
)
l'est puisque
en r@sultera alors par le
th@or@me de convergence domin@e. Pour tout r~el
a> 0
P ~ I [sup
, on a, compte tenu de l'expression donnant ~ ~ n ( ~ ) ~n(
7---
n
F~
CFo {
P
(~)
~ n(F)
e-a
.
n
Lee termes du second membre sont major@s respectivement par
P ~
(F)
et aussi par
-a
e
comme le montre l'argument suivant de th@orie des martingales. Si ~
premier
n
tel que
on a, puisque
P
Sl
P ~ n ( F ) < e-a
{ ~ = n} ~ ~ n
[Fo{~
= n]_~
et si ~
= ®
lorsqu'un tel entier
n
:
= E ~1 t~ P
~n ( F ) . 1 5
- n S~Ze-a P "B'l[ ~ = n ]
par addition des membres extremes de cette in@galit@ il vient :
d@signe le
;
n'existe pas,
- 266 -
P ~1 [ F r l ~ ' , 9
< e::)}'] ~< e - a p '~1 Lr ~ z.
lie ].~ e- a
ce qui constitue l'in~galit~ cherch~e puisque {9 < so} = {inf n
P ~l[sup
~"
P~n(F) 4e'aJ.
n(s)>a
].<
De l'in~galit&
m i n [ P ~'~ (F) , e- a ]
b
que l'on vient ainsi de d~montrer pour tout r&el E
[ Sup ~ ~m( S )] =
fsup ~
da P
n
(S)>
n
a > o , on d&auit par int&gra~±on
a~.~ ~ Fe~
q
_- F~ (~)'------------------~-~ [ P ~ ( F ) zoo- P ~ (F)
:
da min~P ~1 (F), e-aS
(~)
+P
(~)]
L'infigalit~ obtenue montre que : +1.
3) Entropies conditionnelles . D~finition ~ : Etant donn~ deux sous ~ -alg@bres ~t ~'de ~ dams l'espace de probabilit~ on d~finit l'entropie conditionnelle
H ~ (~3')
de la ~ -alg~re
~'
(~,
O~, P )
par rapport ~ d~
en posant :
l z ~ ( ~ ')
EF~m C ~ ' ) ~ .
montrons que cette d&finition coincide avec la d~finition usuelle C ~ J
. Lorsque J~'
est emgemdr~e par ume partition d~nombrable de ~ dams ~ on d~finit usuellememt H ~ ( ~ ' ) comme l'espfirance de
~ p~(d8,) B'an] ( ~ ' )
log [ 1 / P ~ ( ~ ' ) ]
~tant ~gale ~ E ~ C ~ ~
(~')S ; on voit bien que d~ns ce cas
; cette expression H ~(~')
= E[~ ~ (~')B
Pour une sous ~ -alg@bre ~ '
arbitraire, on d~finit usuellement
borne sup~rieure de
lorsque ~ parcourt les sous ~ -Alg~bre de ~ '
H~(S)
H ~(J~') comme la engen-
dr~s par des partitions d~nombrable (ou seulement par des partitions finies) ; il
.
- 267 -
suffit de tenir compte du premier r@sultat du th~or@me ~ pour voir que l'on a encore ') = E [ ~ ( ~ ' ) ~
H~(~
•
Th~or~me 6 : Pour que l'emtropie comditiommelle
H ~ (~')
soit finie, il es~ n@cessaire et
suffisant qu'il existe ~aue ~ -alg@bre ~ engendr@e par une partitiom d@mombrable de ~6 dams~telle
que
a) ~ V ~
=
~
c h o i s i r q de msmi@re ~ ce que
V~'
H(~)~
,
b)
H( ~ ) < ® . En outre om peut toujours
H s(~')
+ 2~Hm(~')
•
D@monstration : 1)
Si 7 est une ~ -alg@bre engemdr@e par une partition d@nombrable telle que
~V7
= $5 V$3' , on a d'apr~s les d@finitione
~(~,) Or
= ~m(~ve,)
H ~( ~)~ H(~)
= ~(~)
e t par suite
E~(~
') =
(~)
.
d'apr~s le th@or@me 7 d@montre ci-dessous.
La comditiom suffisante du th@or@me est ainsi d@montr~e.
2)
Si
H~(~')<~,
la fonction d'incertitude
~ ~ (~Sv~')
finie p.s. ce qui signifie" que la ~ -alg@bre 68 v ~'
est int@grable, donc
est atomlque par rapport ~ ~ .
D'apr~e le th~or@me on peut trouver une partition de d~ en mme suite { Fn, n~l}d'atomes conditionnels de ~ v ~ '
par rapport ~ ~ telle que
Cette d@croissamce de la suite{ P~(Fn),n~> 1~
PS(FI)>~ P ~ (F2) >~ .....
implique que
P~(Fn)L_ 1/n , et on volt
doric que :
7
n~'l
P~(sn) lo~/P~(Fn )~- n7~ l P~(Fn)
log n
et par suite, en prenamt les esp@rances des 2 membres, que Ha(~')>.
~1
P(Fm) log m .
0n ach~ve alors la d@monstration de la condition m~cessaire du th~or@me en utilisamt le lemme suivant. Lemme @ : Si [p n' n >~1} > n>~-I
est ume loi de probabilit@ sur les entiers positifs, on a
Pn log ( 1 / P n ) ~ X + 2 ~
condition de poser
x = 7-n>~l
Pn log n .
- 268 -
D~monstration du lemme : P o u r tout entier
%
a > 0, la s@rie de termes positifs
est de somme inf§rieure ~ 1 puisque
:
De L-in~galit~ classique en th~orie de l'information ~--
Pn log (qn/pn) ~
= exp (-a-~) n -(l+a) ( n ~ )
7-
:
(qn - Pn )4 o
on d~duit alors que :
m•l
Pn l o g ( 1 / p n) < / ~ - P n l o g (1/qm) = a-1 + ( i
E n posant
a = x -1/2
moins si
0< x< ~
avec
1
Pn l o g n .
x = ~-- Pn log n ~ on obtient l'in~galit~ du lemme au n~ I
. Mais dams le cas
n'est r@alis@ que si
+ a) ~ n>
x = co il n'y a rien ~ d~montrer et le cas (n~ 2)
Pl = I ' Pn = 0
x = 0
et l'in&galit~ du lemme est alors ~vi -
dente. Th@or~me 7 : Soit 2 une ~ -alg~bre d e ~ -alg~bre de O n a alors
dans l'espace de probabilit~
engendr~e par une partition
H ~( ~)W H(S)
(~,~,
P )
et soit S une sous
d~nombrable et d'entropie
H(S)
finie.
. De plus les trois conditions suivantes sont alors @qui-
valentes a) b)
les ~ -alg~bres 7 et ~ sont independentes ~ (7)
=
~(~)
p.s.
c) H(~) = H ~ C ~ ) Demonstration
:
Si ~ et ~5 sont ind~pendantes,
on a
P ~ (F) = P(F)
pour tout
F~ ~
et cela entraine d~-
~a l'~galit~ (b). I1 est clair que (b) entraine (c). Enfin comme P(F)
Eta) -H~)
= E(
7 F~ ~
et comme la fonction x = 1
x-l-log x
od elle s'snnulle,
P~(F)[T (~)
P
P(F)
I - log (F)
] ) P ~(F)
est strictement positive sur S o , ® K
on a toujours
H( ~ )~> H 2 ( ~
saul au point
), l'~galit~ n'~tant possible
269
-
que si
P2(F)
s~r ~ P m(F~ > o ~ pour tout
= P(F)
condition @quivaut
PB(F)
l'ind6pendance de
et~
= P(F)
sur~
est proche de
F~w
puisque
(~)
E ~ ~(F)]
; or cette derni~re = P(F), c'est g dire
.
On peut pr@ciser l'@quivalence (b)4=~(c) H(~)
-
H ~(S),
du th@or@me pr@c@dent en montrant que si
les fonctions ~ ( ~ )
et
~(7)
ne peuvent Stre tr@s
diff&rentes. Le th~or~me suivamt @tablit un tel r@sultat avec une certaine uniformit@ suppl@mentaire en ~ .
Th6or~me 8 Soit
"~1 c
(~,
~P
~ 2 c ... )
~ne suite croissante de sous ~ -alg@bres de ~ d a n s
et soit ~ o e
la ~-alg@bre que les ~ n ( n ~ I)
l'espace
engendrent. Soit ~ une
sous g -alg@bre de ~ e n g e n d r @ e par une partition d@nombrable et d'entropie finie. Pour route constante r@elle
a> 0,
on a alors :
n
a+e-a-q
D@~onstration : Soit
G
la fonction r6elle continue d@finie sur
la fonction fonction
G
est strictement positive sur
x G [log(I/x)]
elle admet
1/x
R
par G(x) = ex- 1-x ; notons que
en dehors de l'origine et que la
est strictement convexe sur
= I - x + x log x
R+
fix~ et un ensemble
F de la partition ~ ( ~ ) ,
G[J JSn(~)-
n tel que
G(Iog~P(F) ~ n ( F ) ] ) > b
le premier
~(7)S>bJ=F~(S
pour un r6el
on voit que : )
PfFO{9
F 4
n Or comme I ?F = nJr= '~n : P[F N
{q
F = n
puisqu'
comme d6riv@e seconde .
En d6signant par ~F
Prsup
R
}]
= E[p~n(F). q
{~F
= n}]
1(
P(F)b E [
G[iog
p~o(F
La derni@re in~galit~ r6sultant de la convexit@ de
)
~F
I
=
~F
x G(log ~I ).
n)~
=
'
b>O
- 270 -
En somm~nt cette in~galit6 sur
P[sup G [ ~ S n ( ~ ) n
n ~1
et sur
F@?r(~),
on obtient que :
- ~ ( S ) ] >bS<~-
= ~ E( ~---
K P(F)-P ~
(F)-P ~S® (F) log ~ ] )
=S Pour achever la d@monstration du th@or~me, il reste ~ poser que dams ces conditions
Jxl>a
implique
G(x)>b
(x~R).
b = G(-a)
et ~ remarquer
-
271
-
BIBLIOGRAPHIE
[I]
K. JACOBS s Lecture on ergodic theory. Matematisk Institut, Aarhus Universitet (1962) A. HANEN et J. NEVEU I Atomes oonditionnels d'un espace de ProbabilitY. A
A paraitre aux Acta Math. Hungerioa.
[31
Notions g~n~ralis~es d'incertitude, d'entropie et d'informatioh du point de rue de la th6orie des martingales. Proc. I rst Prague Conf. on Information theory 1957
p. 183-208.
- 272 -
On Markov Processes whose Shift Transformation is ~uasi-m~Ang. Fredos
Papangelou (~0.
The present note is only a sketch of results, with a couple of proofs outlined. A fuller account will be incorporated in smother paper. Throughout, we consider only Markov processes with discrete time parameter and stationary transition probabilities.
~1. Countable state space. Let (pii) be the matrix of transition probabilities of a ~arkcv chain with countable
_(n)
state space I. We denote the
n-step transition probabilities by ~ij "
All chains which we consider in the present note will be assmmed, without further explicit mention, to be irreducible and aperiodic. This is equivalent to the following: For any i,j ~ I
there is no(i,j) such that
_(n) ~ij > 0 for all n i> no(i,j).
Another basic
assumption which must be stipulated at the outset is the existence of a stationary (finite o~ infinite) "distribution" ~j = ~
~i~lj
for every
j ~ I
i
{ ~ il on I;
here 0 4 ~ i ~ +co
and ~- ~i ~ i
+~"
The measure theoretic sample space ( ~ , ~ the present context, the s e t ~
(1)
E
= ~ (Xn)~ ~
corresponding to the given c h A ~
i ~ I,
is, in
of all bilateral sequences of states
.... with the Borel G -field ~
,~)
for every
x I ×I x I × ...
generated by the elementary cylinders, i.e. sets of the form : x r = it, Xr+ 1 = it+ 1,..., x ~
= iv ~
,
~ ~
r
and the measure p determined by the set-function whose values on elementary cylinders such as E are given by (2)
p (E)
= ~ ir
Pirir+l Pir+l ir+2" " ° P ~ -1 i~
( ~ ) The results of this note were obtained while the author was holding a stipend from the Alexander yon Humboldt-Stiftung,
W.Germamy, and had the privilege of many invaluable
discussions with K. Krickeberg at the University of Heidelberg, for which the author expresses hereby his t b ~ s .
- 273 -
The Markov chain itself is described by the shift, i.e. the transformation T of 6 h w h i c h maps each point ( X n ) n a n t o (Xn+l) n.
T is one-to-one, o n t o / h and preserves the measure
-
There is a natural topology in Ih
, namely the product of the discrete topologies on the
individual components I and it is easy to see that this topology derives from a metric which renders Il a Polish space (complete metric space with countable base). Thus, it makes perfectly good sense to ask under what conditions T is mixing or quasi-mixing in the sense of [ 3]. Instead of repeating here the definition of q u a s i - m ~ n g
in its full
generality, we adapt it to the present situation, modifying it slightly for reasons that will be explained elsewhere. Call a subset of6~ and let ~
bounded if it is contained in a finite union of elementary cylinders
denote the ring of all bounded Jordan-Riemann measurable sets in 6h , i.e.
sets whose boundary has p -measure zero. Definition I. The chain (more precisely the shift T) is quasi-mixing if there are two m e a s u r e s ~ l , W 2 on ~
and a sequence
{ 9nl
of positive numbers such that: (i) If
p I(E) < +oo, p 2 C ~ ) ~ +co, and if in addition (ii)
p (E) > 0 then
E 63
then
p i ( E ) > O, ~ 2 ( E ) > O;
for ax~y E,F, E
(3) If p l
lim ~ n ~ ( E n = P2
= ~
T-nF)
=
~i(E)
, T is termed m ~ u g .
It is known that every
~-finite measure on a Polish space is "tight". Using this one
can show that it is sufficient to verify Theorem
(3)
for elementary cylinders. See [ 3~.
1.
The chain is mixing if and only if p(n+m) i"
(~)
~2(F)
14,,
~_)
~j =
for any states i,j,k,h and any integer m. (~) is a so-called strong ratio limit. In its general form, as given by Pruitt in [ 5J, the strong ratio limit property (abbreviated SRLP) is defined as follows.
- 274 -
Defimition 2. ~(i)
The c h ~
(i c I)
(5)
is said to have the SRLP if there are positive numbers ~ , T (i),
such that
lim
~(n+m) ~i~
n~@
p
for any states i,j,k,h
m
~(i~(j)
and any integer m.
Theo!em 2. The chain is q u a s i - m ~ n g
if and only if it has the SRLP and the following equalities
hold (6)
~T(i)
= ~.
pij~(j)
(7)
~(j)
= ~(i)
i
for every i a I for every j a I
Pij
under these conditions, if E is given by
(I)
then
Cs)
HICz ) : f-u ~i r Pirir+ I ....p~_liv~Ci~ )
(9)
H2(E)
: ~r~(ir)Pirir+1
.... R ~ _ l i 9
For the "if" part of th.2 cf.[3] • The import of (6) and (7) is to insure Kolmogorov's compatibility conditions for ~ I ' V
Pij ~CJ)
and ~ - C j )
~2"
~ ~Ci)i
It is easy to see that Pij"
(5) implies ~ ~ (i)
One may have strict inequality ; there is
a model of a Markov chain such that the SRLP implies (6) and (7) if and only if the chain is R-recurrent
(see definition below). However this model admits a stationary "distri-
bution" only if it is reourrent. From (5) we see that ~ is the convergence norm of the chain (Vere-JonesE7J), i.e. the reciprocal of the radius of convergence R of the power series p(n)
xn. Clearly
R~> I.
Vere-Jones has shown ([7])
that R is independent of i,j, and that we either have
In the former case
(n) Pij
Rn = + ~ for all i,j
(n) Pi~
Rn<
+~
or
for all i,j.
he called the chain R-recurrent and in the latter R-transient. He
further proved that in the case of R-recurrenOe equations (6) and (7) have unique
(to
- 275 -
w i t b i ~ a comste~t f a c t o r ) p o s i t i v e solutions
f~(i)}
,t~TCi)}
and that there exists
a non-negative number c o with
(lO)
~(n) H n
lira
=
Co~:(i)~rCj)
for all
i,S •
The chain is called H-positive or H-null according as c o > 0 or
=0. If the chain is
H-positive we therefore have the SRLP and quasi-mixing. If it is R-null we may or may not have the SRLPI fir (i)
but once we have it, (5) is satisfied with the same n u m b e r s T ( i ) ,
that constitute solutions of (6) and (7).
Theorem 7If the chain is quasi-mixing and { ~ ( i ) 1 is the unique
(to within a constant factor)
positive solution of (6), then the measure p 1 is either absolutely continuous or singular relative to p Sketch of proof.
. If
~ =1,
Decompose
then
~1
= ~
"
P l into absolutely continuous
and singular part
•
pl(A) = /fCx) F (ax) + e(A) A
Differentiating every
Pl
relative t o p
x = (...,Xl,Xo,Xl,...)
(11)
f(x)--
on the obvious net in ~
we find that for a~most
in
lira
Z -n ~:(xn)
n--A oo
I f ~(i) = E ( f I x o = i), then {~(i)} such that
~ (i)
to ~ . ~If ~ )
= ~-(i) 0, then
is a solution of (6). Hence there is a constant~.~0
for every i ~ I. If ~ = 0, ~ is obviously singular relative ~(i)
~-n~(x n)
= I/~ E( f l Xo=i)
=
y~
ECfl % ,
and one can show
Xl,...,x n)
w h i c h m e a m m that in (11) we have a uniformly integrable martingale (on each ~ i
= {(xm) If [ =
: x ° = i } ). This implies 1, then
OL = 1 and
Q(~q) = O, i . e .
HI(A)
=
= A~f(x) H(dx).
T ( i ) = 1 for every i, by the uniqueness of the solution.
E q u a t i o n (7) is similarly Connected with ~ 2" To show this, it is sufficient to consider the "inverted" Markov chain. Note that both (6) and (7) have unique solutions if the chain is R-recurrent.
- 276 -
Theorem ~. If the chain is R-positive and R > I (i.e. ~ < 1), rive to ~ Infact,
then
~I and
P2
are singular rela-
.
if An = I x : x o = O, x n = O}
then~
-n~(~n)H(~
This means that the sequence
)
(An) = ZoPoo ^ (n) --+ 0
while
(n)_. ~o~(O) ~(O)Co~(O) > o = ~o~(O) ~n Poo
~ - n ~ ( x n) in (11) is not uniformly integrable o n 6 ~ o.
2. Markov chains with independent increments. Consider an irreducible, tegers
(Pi,j
aperiodic random walk with independent increments on the in-
= P o , j - i )" I t admits t h e s t a t i o n a x . y " d i s t r i b u t i o n "
Chung and Erd~s (~I]) proved in q951 that if lim n~co
~=q,
)i
= 1
(i • I).
then
(n+m) ~ij (n)
=
I.
Pith Kemeny later generalized this result as follows ([2]). Let +~ f(s) = 7-Poi sl , 0 ~ s ~ +co. i=-co i The chain has the SRLP, with F = f(So) , T(i) So, = ~ ( i )
= s; l, where s o is the
unique positive number with 0 < f(s o) = inf f(s) ~ I.
s Equalities
+~ and (7) all collapse to f(So) = 7--
(6)
Poi s io
which is true.
i= --GD Theorem ~. If
~ = I the random walk is mixing. If ~ < I, it is quasi-mixing with ~ I ,
~2
singular
relative to ~ . Sketch of proof.
Let~l.
From(11)
Xn so
d~ 1 (x)
=
lim
n~+co
~-n~(xn)
=
lira
n-~+co
f(so)n
=
lira
n-*+co
Xn/n so (f(So) )
n
By the strong law of large numbers Xn/n converges for almost every sample point to the "expectation"
of the increment,
which may be +co or -co, and can easily be seen to be
- 277 -
f'(1). One concludes the proof by showing that
equal to
olf f(so) §~. Continuous state space. We shall only discuss the following case: A process with discrete time parameter and independent increments on a locally compact Abelian group G. We denote its tr~naition probabilities by p(x,A)
(x ~ G, A a Borel subset of G). Our sample space is
....
×G×GxGx
...
with the product topology. The strong ratio limit theorem of Chung-ErdSs and Kemeny was recently generalized by 0rnstein ([$S) and C. Stone([6S).
We present here part of Stone's general result.
Following Stone we assume that (i) G is compactly generated,
(ii) p(O,.) is a regular
Borel measure on G and (iii) the closed semigroup generated by the support of p(O,.) is G itself. Let
~ be the Haar measure on G. It is a stationary measure for the process. Denote by the collection of all continuous homomorphisms of G into R (the real line) and de-
fine g(s)
=
J
e s(x) p(O,dx)
s a
G
let ~
Finally,
be the class of all bounded Jordan-Riems~n measurable subsets of G which
have positive Haar measure. Stone (E6J) proved that there is a unique 0 < g(s o)
=
inf
g(s) ~ 1
and that if
Aa
~,
B E ~
then
s~ lim n~oo
p(n+m)(xoA) p(n) (y,B)
=
g(so)m
so ~
~
e_So(Z)
~
~ (dz) e-s°(z) ~ (dz)
eSo(X-y)
such that
uniformly with respect to x and y in compact sets. Using this result one can prove: Theorem 6. If So~ O, then we have mixing. If s o is not identically zero, then we have quasimixing, with the values of ~1' E
=~
x
:
x r ~Ar,
~2
on elementary cylinders of the form
Xr+ 1 ~ At+ 1,..., x v ~ A
}
- 278 given by
r ~2(E)
= g(so) r
r+1
~Ar e-S°(Xr)~(dXr)/Ar+l P(Xr'dXr+l)'''~A P(Xv-l'dXw)
If G = R, then ~ can be identified with R and one can prove law of large numbers): Theorem 7. If
so { O, then y1' ~ 2
are singular relative to ~ .
(using again the stx~mg
- 279 -
References. (I)
K.L.Chun~, P.Erd8s, Probability limit theorems assuming only the first noaent. Memoirs Amer. Math. Soc. No. 6
(2)
(1951).
J.G. Kemeny, A probability limit theorem requiring no moments,
Prec. Amer° Math.
Soc. 10, 607 - 12 (1959). (3)
K. Krickeberg, Strong m ~ i n g
properties of Markov chains with infinite invariant
measure, Prec. Fifth Berk. Symp. on Prob. and Star. t to appear. (4)
D. Ornstein, A limit theorem for independent random variables, unpublished.
(5)
W.E. Pruitt, Strong ratio limit pzoperty for R-recurrent Mamkov chains, Prec. Amer. Math. Soc., 16, 196 - 200
(1965).
(6)
C. Stone, Ratio limit theorems for random walks on groups, to appear.
(7)
D. Vere-Jones, Geometric ergodicity in denumerable Marker chains, Quarterly J. Math. Oxford, Set. 2, 13 ,
7 - 28
(1962).
- 280 -
Remarks on the Poisson process. A. R@nyi
The (inhomogeneous) Poisson process on the real line is usually characterised stochastic
additive set function
as a
~ (E) defined for each bounded Borel subset E of the
real line such that a) the random variable
# (E) has for each bounded Borel set E a Poisson distribution,
i.e.
[~(E~n (I)
= n)
• e- Z(E)
=
(n=
o,I,...
)
nl where
~ (E) is a nonatomic measure on the real line such that
finite interval b)
E,
If we put
~ (E n)
are mutually disjoint bounded Borel sets the random variables are independent.
~t =
~ ([0,t))
dent increments
such that
A(t)
- A(s)
-A(t)
is the
(E) is finite for each
and
if El, E2,...,E n (E 1),...,
~
where
for t > 0, this means that ~t -
A (t)
~s
~ t is a process with indepen-
has a Poisson distribution with mean value
is the k -measu-~e of the interval
~ -measure of the interval
It,0) if t 4 0.
[0,t) if
t > 0 and
D.Sz&sz (oral communica-
tion) asked the question whether there exists a point process for which a) holds but b) does not hold. We shall show in this note that such a process does not exist, i.e. the usual supposition about independence cessary,
in the above characterisation
of the Poisson process is 1 ~ e -
as it follows from the Poissonity of the distribution of ~ (E); in other words
we prove that the supposition b) is a consequence
of the supposition a).
More exactly we prove the following Theorem 1. Let ~ denote the family of all subsets of the real line which can be obtained as the union of a finite number of disjoint finite intervals [a,b) open to the left. Let E ~ ~
~ (E) be an additive stochastic
, i.e. such that if
closed to the right and
set function defined for each
E I and E 2 are disjoint one has
~(EI+ E2) =
~(EI)
+~(E2)
-
Suppose that for each E ~ ~
281
-
~(E) has a Boisson distribution with mean value
A (E) where
(E) is a nonatomic measure on the Borel subsets of the real line, which is finite for each E • ~ variables
. Then it follows that if El, ...,E n are disjoint sets J(EI) , ...,
Proof of theorem q. disjoint sets
](En) are independent,
Let
Ej e
i.e.
~ (E) is a Poisson process.
A(E) denote the event ] (E) = O.
(j = 1,2,...,n)
( E k ~ ~ ) the random
then (~) clearly
If E is the union of the
A(E) = A(E I) ... A(E n) be-
n
7-
cause
(Es) and thus
] (E) = 0 iff ~(Ej) = 0 for
S = 1,2,...,n
S=I But by supposition n
(2)
n
P(A(E)) : P(~ (E) = 0) = e- Z(E)
e-Z(Ej)
=
j=q Thus it follows that if the sets El, ... , E n A(En)
U
P(A(Ej))
j=1 are disjoint, the events A(EI) , ... ,
are independent.
Now let IA(E) be the indicator of the event A(E). Let E g ~ sets. For any
and F a ~ b e
two disjoint
E > 0 we can clearly decompose E into disjoint intervals E i
(I g i~ n)
and F into disjoint intervals such that max i
~ (E i) ~
£
and max A(Fj) j
< 6
n
NOW evidently
~ (E)
~
~
IA(Ei)
implies
(A(Fj)
implies
= m
and
~ (F)
~
max i
~ (E i) % 2
I
0~I=
max
~ (F j) .~ 2.
J On the other hand for any
B ~ ao
(3)
P(~ (B) ~ 2)
=
Z(B)~e
~-k=2
- Z(B)
%
~2(B)
k!
Thus (~a)
n
P(~ (E) { ~
n
IA(Ei)) % ~
l~(Ei)
< £ }k(E)
i=I
(~) Here and in what follows the product of events denotes the joint occurrence of these events.
- 282 -
and
m
m
~--
=
S=1
2(Fj)
This implies, as the sums IA(E i) are independent, that
m ~=
and
~ (E)
and
~(F)
As a matter of fact it follows from
1ACFj) are independent too.
(4a)
and (~b)
that for any n and m
(n,m = 0,1,
2,...)
(5) As
£ > 0 can
be chosen arbitrarly small, our statement follows. The independence of
the variables
~ (E i)
(i = 1,2, .. ,r) with disjoint E i and r > 2 is proved in exactly
the same way. Thus our theorem is proved. Remark1. if i ~
Note that to prove the independence of
(E i)
j we have not used the full supposition
(i:1,2, .. , r)
that for each E ¢ ~
for EiEj--~
~ (E) has a
Poisson distribution, only that
and
(6a)
P ( ~ (E) = 0 )
(6b)
P(~ CE) ~ 2) =
e-
=
A(E)
G(Z CE))
if
Z CE)
--*
0
uniformly in E. Thus even these suppositions imply that the process
~ (E)
is a process of independent
increments. It is easy to show however that this together with (6a) and (6b) implies that
~ (E) has a Poisson distribution.
Thus the following theorem is true. Theorem 2. Let ~
denote the f~m~ly of all subsets of the real line which can be obtained as the
union of a finite number of disjoint finite intervals [a,b). Let stochastic set function defined for disjoint one has
~ ( E 1 + E 2) =
E ~ ~
) (E1) +
~ (E) be an additive
, i.e. such that if E 1 ~ ~
~(E2) . Suppose that
and E 2~ J
~ (E) is for each
are
- 283 -
a non-negative integer valued random variable such that (?a)
P(~ (E) = 0)
=
e - ~ (E)
and
such that lim ~ (x)=0 x-~O (E) is a Poisson process,
where ~ (x) is an increasing positive function defihed for x > 0 and Z (E) a nonatomic measure on ~ . Then i.e. if
Ei
(i=1,2, ... , r)
it follows that
are disjoint sets,
Ei ~
the random variables ~ _ (i) E
(i = 1,2, .. ,r) are independent, and (1) holds. Proof of theorem 2.
~E(U )
(8)
Put for E s
= M(eiU ~(E))
J(,jPE(u)/ _a. e - ~ ( E )
( -co< u <+co)
- I ut(
1 - e-'~CE)
)
>
then clearly
0
if
1 )u~<
e Z(E)-I
Thus if
r
E
=
k i=1
lu I <
e
E i,
where
I ~(E)_I
E ig
--a
EiE j
=
if i ~ j, then for
we have
r (9)
~(u)
TV
=
,cu) i
i=1
o
and therefore
r
(qo)
log ~(u)
: ~_
log WEi(u)"
As however
i=1
(11)
(12)
~Ei(U)
=
log ~w.i(u)
It follows that if (13)
lOg~E(U)
e-/~(Ei)
=
+ eiu (1 - e - A(Ei)) + 0 ( Z ( E i) S ()~(Ei))
~ (Ei)(eiU - 1) + 0 ( ~ (Ei)(~ (E i) + 8 ( Z (El)))
~ ( ~ i ) ~ E for i = 1,2, ... , r =
~(E)(e iu-
I) + 0(6
+ ~(~))
we get
--
284
-
that is, as g > 0 can be chosen arbitrarily small,
(14)
WE(u)
e Z(E)
:
(eiU - 1)
w h i c h implies that ~ (E) has a Poisson distribution with mean
(E). Thus theorem 2
follows from theorem 1. Remark 2.
The proof can be carried over without any change to the discussion of a
Poisson process in more than one dimension or even in an abstract space. Thus we obtain the following Theorem ~. Let ~
be any space,
a family of subsets of ~ a n d
~ (E) a non-negative finite va-
lued set function defined on ~ , such that I)
if
E 1~7,
E2 ~ ~
2)
E le~
3)
There is a constant % with 0 < ~ <
, E2¢~
and EIE 2
, EIE 2
=
=
~ , then
~ then
E I + E2 £
Z ( E 1 + E 2) I
<
1 -
Z ( E 1) +
Z(E2)
such that for every E ~ ~
there exists a subset F of E such that F c ~ , < / ~
=
E - F¢
~
with
~(E) >
and
oC
Let us s~ppose a stochastic set function is defined on ~ , i.e. to every there corresponds a random variable ) (E) EIE 2
=
~
we have
0
~ (E 1 + E2)
distribution with mean value
=
such that if E 1 ~ ~ , E 2 ~ ~
~ ( E 1) +
) ( E 2)
and
~ (E). Then the random variables
... , r) are independent if the sets
E ig ~
(i = 1,2,
E e and
~ ( E ) has a Poisson ~ (Ei)
(i=1,2,.
..., r) are disjoint,
i.e.~ (E) is Poisson-process. Note that condition 3) is not quite the same as that ~ is nonatomic, because we did not suppose that ~ is a Remark ~.
~ -algebra of sets.
The question arises whether the condition that the process should be one with in-
dependent increments can be deduced from other suppositions for other processes of independent increments too. The most i n t e r e s t i n g case is that of the Wiener process; for this process one has the following (almost trivial) analogue of theorem 1.
- 285 -
Theorem 4. Let
~t (-co ~ t < +co) be a stochastic process such that ~ t - ~s is normally distribu-
ted with mean 0 and variance c(t - s) intervals
[ sj,tj)
(c > O)
for s< t. Suppose further that if the
(j = 1,2, ..., r) are disjoint, any linear combination r ~=1 bj( ~ tj - ~Sj ) of the increments ~ tj - ~ sj t
~
coefficients b l is normally distributed. Then ~ t random variables
~ tj
-
~ sj
with real
J
is the Wiener process, i.e. the
are independent if the intervals
[ sj, tj) are
disjoint. Proof of theorem ~.
Clearly putting for I k = ~ sk,tk)
if 11 and 12 are adjacent intervals
~ ( I k ) = ~ t k - ~ s k (k=1, 2) (I1 + 12) = ~ ( I I )
(s1 < t I = s 2 ~ t 2 )
+ ~(I2)
and thus
M((~ ( I 1 + I 2 ) ) 2) = t 2 -s I = t 2 - s 2 + t 1 - s I =
= M(~2(I1)) and thus
M( .~ (I1) ~ (I2))
let 11 and 12
and put
+ M(~ 2 ( I 2 ) ) =
0
, i.e.
~ (I I)
and
~ (I2)
are uncorrelated. No.
be arbitrary disjoint intervals
11 : [ s1,t 1),
12 =
I3 = Etl,s2) .
Then, taking into account that
M(~ (I1)~ (I3))
fs 2,t 2)
=
0
and
where
slz t I< s2~ t 2
M ( ~ ( I 3 ) ~ (I2))
~ ( ~ 2 ( I I + 12 + I3)) = t 2 - s I : M(~2(II))
=
0 ,
we get
+ M ( ~ 2 ( I 2 )) + M ( ~ 2 ( I 3) +
+ 2M(~ (I1) ~ (12))
~us
~() (II)) (z2))
:
o
(We have used here the following elementary geometrical fact: if a,b,c are vectors in the 3-dimensional Euclidean space for which c is orthogonal both to b and to a + b, then c is orthogonal to a too.) Thus
~ (I 1) and
~ (I2) are uncorrelated if 11 and 12 are arbitrary disjoint i~tervals.
It follows that if 11,I2, ..., I r
are disjoint intervals and Ij has length I Ij I ,
and b I , ..., b r are arbitraz~z real constants then
- 286 -
r
b~ clzjl j;'l
j=1
r
r
I~2 Z 2 - ~ S=I bS c Izjl
JuTbj~ (lj) j=l M( e )
Thus
= e
and thus for any real numbers Ul, u2, ..., u r
r
i SZI= uj ~ (lj) M( e
r )
= ~
iuj)(zj) M(e
)
S=1 i.e.
the
(Note that
] (Ij) it
(j = 1,2, ..., r)
would h a v e b e e n s u f f i c i e n t :
Remark @.
are independent, i.e. 5 t is the Wiener-process. to suppose that
bj ~ (lj) is normally distributed for
~=
b
~1.)
Returning to the Poisson-process, the question arises, whether if in
theorem 1 instead of the condition that # (E) has a Poisson distribution if E is any finite union of intervals, one supposes only that ~ (I) has a Poisson distribution if I is any interval, does this still ensure that the process is a Poisson process? It is easy to show that in this case
~ (I1)
and ~ (I2) are uncorrelated if 11 and 12 are
disjoint intervals. The proof of this is essentially the same as the first step of the proof of theorem @.
Remark added on August 22, 1966: I have been informed by Jay Goldman, that the answer to the question in Remark 4 is- no. This has been shown by a counterexample by L.Shepp; this example will be published in a forthcoming paper of J.Goldman.
- 287 -
Sums of Markov-chains
on finite
semigroups.
L. Schmetterer.
Let S be a finite set.Suppose written as a multiplication.
that a binary operation is defined in S which shall be Let
( ] n)n ~ 1
be a homogeneous
Markov-chain
whose state
space is S with transition matrix P = (Pgk)g,k¢ S" Let us consider a new stochastic + process which is defined by s n = ~1 ~2"'" ~n' n ~ 1, and which is called the sequence of right sums of the chain ( ~ n)n > 1 .The problem arises to determine viour of
s n+ .
A similar problem is concerned with
quence of left sums of the chain interesting complete
s~
= In
~1'
n ~l,the
( ~ n)n > 1 " It seems to be very unlikely
answer to this problem exists under such general
answer is available
"'" ] 2
the limit beha-
if the operation
assumptions.
defined in S is associative
se-
that any But a more or less that is:
S is a semigroup. If S is a finite or even compact group and if ( ~ n ) n > I is a sequence identically
distributed random variables it is very well known that the possible limit + of s n are the Haar measures on compact sub-groups. If S is a discrete (not
distributions necessarily tically
finite)
semigroup
distributed
Per Martin-L6f
~1]
F o r a (general) finite
of independent
and if ( ~ n ) n ~ I is still a sequence
random variables
of independent
related results may be found in a recent paper by
.
homogeneous
semigroup no results
Markov-chain
(~ n)n.> I
with state space S where S is a
in this direction have been published
so far as I know. The
method which I am going to use is simple and in a certain sense purely algebraic. also [2~
iden-
Cf.
.
Let me introduce
a few notations:
We write p m = (p(~))
. It is said that k ¢ S
may be
reached from g e S if there exists an m > 0 such p(g~)> 0. This is denoted b y g ~ k . Obviously
the relation
~
is transitive. The state space S being finite
cation of the states of ( ~ n ) n ~ l unessential. implies k / ~ g . g~
is as follows:
Let us recall the definitions: g is called unessential
k but not k ~
g. An essential
A state g ¢ S
the classifi-
is either essential
A state g is called essential
or
if g ~ k
if there exists at least one k e S such that
state is either positive
recurrent
or recurrent-pe-
- 288 -
riodical. A n unessential state is transient. C ~ S is called a closed class of (essential) states if each state of C can be reached from every state in C and only these states can be reached from the states in C. Unfortunately (s+) n ~ q is in general not a Markov-chain but the process S+
whose state space is S × S is a Markov-chain. Therefore we consider the homogeneous Markov-chain
(~n)n>~l
whose state space is S ~ S
and whose transitioumatrix is given
by
sl w(li+
1
---
: J o
sI
sk
) II i Pgk
Sl = sk +
To study the possible limit-distribution of ( s )
n n ~ 1
leads to the problem to classify
the states of -(~n)n -~ ~ I if the classification of the states of "D (~ n)n- ~ 1 is given. We proceed with the almost obvious Lento- 1:
Let g be a transient state ( ~ n) n >I 1 " Then each state (g), s ~ S of (~n)n~l
is trsn~ient. We may therefore restrict ourself to the classification of such states (~) where g is essential.~e may even assume that S is a closed class of essential states. This is not an essential restriction. Let us point out that the case where S (or any other closed class) contains a zero is trivial: Lemma 2: g~S
Let S be a closed class containing a zero ~. Then exactly the states (g),
of ( ~ n ) n ~ > l
are essential.
Obviously it follows from g ~ k
and k ~ g
for some g,k¢ S that (~)~(~l)_ = ( ~ ) G ( g
1)=(g)
for some l, 11 a S and one part of the lemma is proved. If s ~ o then it follows from gr~e that ( ) f ~ (
~2 ) = ( D
for some 1 2 a S , that is: ( ) is transient.
From now on we may assume that S does not contain a zero. Next we mention a simple but important i emma. Lemma ~a
Suppose that g e S is an essential state of (~ n)n~> 1" Let
(s) be an arbi-
- 289 -
-trary state of
( ~ n)n >~1" The set Hg of all elements h which satisfy (g) ~
(gh)- is a
semigroup which is called the invariance semigroup of g. Now we are going to formulate the Theorem 1 : Let S be an abelian semigroup of order ~ . Let H be the universal note the order of H by ~ • Let the state spaces of (i n)n>. I the states (g), s g H ,
gES
of ( ~ m ) n ) 1
(twosided)
ideal. De-
be a closed class. Then
and only these states are essential. Let e be
the unit element of H: Let Hg be as above. The sets eHg are identical for all g~ S and may be denoted by the same symbol H~. Furthermore, The ~
states (g), s ¢ H, g ~ S split in
H ~ is a subgroup of H of order v say.
~ / v closed classes ea=h of them containing
v~ elements. It is easy to show that theorem 1 may be generalized to non-abelian
semigroups
S which
contain a group-ideal H. The only difference is that the set eHg are not identical but are conjugate
subgroups
of H. The group eHg may be called invariance group.
It is possible to complete the theorem 1 by the following Theorem 2: Let S be a (nonabelian)
semigroup S. Suppose that there exists a group-ideal H of order ~7 • Let S be
a closed class. Assume that the states of S possess period d >~ 1: Let v be the order of any group eHg. Then all the
~ / v closed classes of (~n)n>~ 1 have the same period
which is the form m d, 1 < m ~p. The period can be different from d o r ~ the group eHg possesses a non-trivial g
d only then if
invariant subgroup or more precisely:~ Choose any
S. Then there exists an uniquely determined invariant subgroup H~1)~ of eHg (which
may be equal to { e } or equal to eHg)
such that m = v/v (1)
where v (1) is the order of
H~g 1)," The groups H~g 1)" and --~l)are conjugate for arbitrary g, k ~ S. They may be called the period-groups. Using well known facts about the kernel of an arbitrary (finite)
semigroup it is pos-
sible to generalize theorem I and 2 to this case. We are not going in details here and proceed to some limit theorems. Suppose that S contains only one closed class C and a class of transient states which may be empty. Then it is very well known that N
always exists and is equal to Q = (qgk)g,k a B say, where
qgk = qk'
g,k e S. goreover
-
the relation QP = P Q = Q = Q2
290
-
holds. Vice versa Q is uniquely determined by this rela-
tion. This leads ~ogether with theorem 1 to the following Theorem ~: Suppose that S satisfies the (algebraic) assumptions of theorem 2. Furthermore assume that'S contains only one closed class C amd a class of transient states which m a y b e empty. Let of
CnH
~ ~. Let eHk, k ¢ C
be the invariance group (of k) and v the order
(which does not depend on k). For every s c H n and all Sl, g, gl e S the rela-
tion
N 1
9i=1
N-~
s
holds whenever
s
(g~) ~ (g)
qg
is satisfied.
Let us point out an important consequence of theorem 3 which we formulate as Theorem 4s Suppose that S satisfies the (algebraic) assumptions of theorem 2. Let S be a closed class of positive recurrent states.Whenever the equation
SlX=
s2, S l , S 2 ~ S has a so-
lution x ~ S then
lim W(s~ = slsl,s2) exists for every s E H and is the H a a r measure on n*oo H iff the invariance group and the period group are equal to H. F o r more details and a fuller account see a forthcoming paper in the Monatshefte fGr Mathematik 1966.
References.
(I)
P e r Martin-L6f,
Z.Wahrscheinlichkeitstheorie
(2)
J.Ciglerund
Trans.Third Prague Conference,
Schmetterer
L.
verw. Gebiete 4, 78 - 102
(1965).
Information Theory, Statist.
Decision Functions, Random Processes, Prague 1964, 45 - 53.
-
291
-
On superefficiency. L.Schmetterer
Let (R,S) be a measure space and Pt a probability measure for each t g T where T is for the sake of simplicity an open subset of one-dimensional euclidean space ~ .
Suppose
that the measures Pt are mutually absolutely continuous. Denote the space of all Pt-integrable functions by ~ t "
Let ~ be any real-valued function defined on T. Further-
more assume that there exists an S-measurable h from R to ~
which belongs to
~ tgT
t
such that
E(h,t)
I
=
hap t
for every
~(t)
=
t ~ T.
Denote the set {h: h g f ~ t~T The norm of the space ~ t = Let
l~t , E(h,t)
will be denoted by
mh(t) as h goes through H ~ . &Pt ~to
t O a T and denote
~(t),
=
by HV: .
11. IIt • Let us consider IIh - W(t) Itt =
Define m(t) = oo if by f (t°) •
t a T1
h~t
Suppose that
° f (t°) ~ ~ t O
for all t ~ T
and every fixed t o. Under these assumptions the following result holds which has been frequently used in numerical analysis and in methematical statistics:
For each t e T there exists min mh(t) = s(t) where it is understood that s(t) = co if haH HW n ~t = #" It is very easy to prove this. Suppose that for some t o e T the intersection H ~ t ~ o Denote the linear closed subspac@ of ~ 2t °
s p ~ = e d by all ft (t°) by ~ .
h ~ H~ ~ ~
and h o the projection of h on72,. Then s(t o) =l~h o to suitable regularity conditions it is even true that
(1)
Let
W(to ) /]to. Under
1 s2(to )
~
where
z(t o)
T(t o)
=
E(
~t
~ t°)
~ow, let h ¢ H v and assume that mh(t ) is finite. Let {hnl be a sequence of indepen-
@.
-
292 -
dent identically distributed random variables which have for each t e T the same distribution (under Pt ) as h. It is well known that the sequencelh(n)lwhere
h (n) = (hl+...
. ..+hn)/n converges in Pt-probability to ~ ( t ) and that ~ ( h (n) - ~(t)) converges under Pt in law to N(0,~(t)). It has been believed for a long period that even the following statement is true: Let {(R (n)
s(n)) 1
probability measure on (R (n), S (n))
be a sequence of measure spaces. Let P(tn) for n.~ 1 and each t~ T. Let h (n)
random variable on (R (n), S (n), P(tn.) ), u P 1
and suppose that ~ ( h (n)
be
be an arbitrary - vJ(t)) conver-
ges under P(~) in law to
N (0, ~ 2
Denote the set of all sequences I h(n) I in~
2 6- lh(n)j(t)
lh(n)j(t)),
> o.
which satisfy these conditions by H. Then
~ 2
(t) > 0.
fh (n)] ~ E
~'hCn)j
If P(~) is an n-fold product measure and some obvious regularity conditions are fulfilled then
~ ~h(n)] (t)
satisfies an inequality analogous to (1) whatever
~h n } e H
is. This statement has been first disproved by J.L. Hodges jr. The following result [I], [2] is a little more general: Let T be an open subset of R 1 and let T o be a countable compact subset of T. be a random variable on (R (n) , S (n) , P(~) ) , plicity
W (t) by t and suppose that
T,et h (n)
t a T , n ~ l . Replace for the sake of sin-
V-m(h(n)
-
t ) converges (under P(~)) in law
to N (0, O 2 (t)), t a T. Assume that
sup ~ 2(t) is finite. Then there exists for ta T (n) of random variables such that each real /3, 0 < /3 < I a sequence {h J ~ ( h (n) - t) converges (under P(~)) in law to N(0, ~ ( t ) ~ 2 ( t ) ) where
~(t)
=
I, t~m-m o
and ~ t ) ~
, t~m o.
On the other hand there exist important examples of classes C of sequences { h (n) } such that ~ ( h (n) - t) converges in law to N (0, ~2 inf {h(n)}~ C
6-
2
(t))
(t) ~ I h(n)~
I
and such that ,
t ~ T.
I(t)
The following theorem is concerned with this fa-~t. Theorem I : [3], [4], [5]. Let ~ be a random variable on ( ~ , ~ l , P t ) ,
a
t e T where T is an open
- 293 -
subset of R I
and ~ l i s the ~ -algebra of all one-dimensional Borel sets. Suppose that
the measures Pt are dominated by a 6~-finite measure. Denote for every t ~ T the corresponding densities by f(.,t). Let us assume that
~[(1og f( ~ ,t) - Zog f(S ,to)); to J =
-I/2 (t-to)2 [ I(to) + o(1)]
and
[(log f(~ ,t) - log f(~ ,to))2;to ] =
(t-to)2 [I(to) + o(I)]
locally uniformly for every t o c T.~) Furthermore, t -*E((
~log f(f ,t)- )2 ; t) =
assume that
z(t)
Bt exists, is > 0 and is continuous. Let ~1' ~2'''"
be a sequence of independent identi-
cally distributed random variables with the same distribution as S . Suppose that h (n) is defined on euclidean
R n and is Borel-measurable,
converges in law to N (0, ~2(t)) where
n.>1. Assume that
[ (n) = (~1''''' ~ n )"
~-d(h(n)o ~(n)-t)
Let us denote by Fn(.,t)
the distribution function of ~(h(n)o ~ (n) _ t) and assume that Fn(Y,t) converges continuously to 0 ( y /~(t))
for each fixed y E ~
function of the normal distribution.
and t a t
where @ is the distribution
Then the inequality ~2(t)
~> I/I(t)
is true for
all t ~ T. Corollary:
If continuous convergence is not assumed then
in a set of Lebesgue-measure
~2(t) < 1/I(t)
holds at most
O, the so-called set of superefficiency.
The proof of theorem I rests on the following lemmas: Lemma I: ~],
[5J. Let t n = t o + 1/V~, n.>1, t o a T .
Then
n
T[f(~i,tn) W~O
I(l°g i=In
+ I" I(t°)~
/
(I(t°))'/2
~
Y~"~'(Y)
We( [i,to) i=I and
~) I take this opportunity to point out that the corresponding uniformity-condition [5], p. 305 has not been stated.
in
- 294 -
n W ~n
og i=ln
I T f(
i=1
+ 1/2 I ( t o )
( I ( t o ) ) 1/2 g
-~
i, to) --~
Lemma 2:
y
~ ( y - (Z(to))Y2).
(Neyman-Pearson). Let PI' P2 be two (different) measures on a measure space
(R,S). Let fl resp. f2 be densities of P1 resp. P2 relative to some dominant measure. Let N be an arbitrary set a
P2 {
x:
f2(x) ~ kf l ( x ) J
.
This
S and k ~ 0 some real number. Suppose that P2(N)
implies
PI(N) > P1
{x:
f2(x) > k f l ( x ) j
.
A proof of the corollary uses the following lemma Lemm~ ~:
~4], [5] :
Let fn be a sequence of Borel-measurable functions on R 1 such
that fn-*O holds almost everywhere (in the sense of Lebesgue). Let B(K) be the set of all sequences {ykl where l ykI~K , k ~ l .
K > 0 and let
Then there exists a subse-
quence ~nk] of natural numbers such that fnk(x + Ynk ) converges pointwise to 0 whatever ynl~ B(K)
is~up to a set of Lebesgue-measure 0 which may depend on the sequence ~ y n ~.
When the sequence ~ ( h ( n ) o
~ (n) _ t))
converges in law but not necessarily to a
normal distribution then a related theorem may be proved: Theorem 2: E5J, C6~ : suppose that the following assumptions of theorem 1 are satisfied with the following modification: Fn(.,t) converges continuously to a distribution function F(.,t) for every fixed ta T and Fn(Y,. ) converges continuously to F(y,.) for every fixed y ¢~.
Then there exists l(t) = inf[ y: F(y,t) = I~ } and u(t) = sup {y: F(y,t) = I/2 2
for every t ¢ T. Let V n be defined on R n and supposa that V n is Borel-measurable. Assume thatIVn (V n - t)) lim Wt(- ~ n~oo
converges in law to
N (0,1/l(t)). Then the inequality
< ~ ( V n O ~(n) - t) 4 ~ 2 ) ~
lim W t ( - ~ l + l ( t ) < ~ ( h ( n ) e n~
~(n) _ t ) ~ 2 + u ( t )
holds for each t ~ T and arbitrary positive n u m b e r s 4 , ~ 2. If continuous convergence of the sequence F(y,.) is not assumed then the inequality is violated at most in a set of
Lebesgue-measure 0.
- 295 -
Theorem I may be generalized to multidimensional open also been extended to Markov-chains
~1'
~ 2' "'"
sets T C3J • Both theorems have
[5]
References.
(1)
L.Le Cam,
(2)
L.Schmetterer, Einftthrung in die Mathematische Statistik, 2. Auflage, Springer-
Univ. California Publ.Statist. 1. 277 - 330, 1953.
Verlag: Wien 1966, 413 - 415.
(3)
C.R. Rao, Sankhya, Set. A, 25, 189-206 (1963).
(4)
R.R. Bahadur, Ann. math. Statistics 35, ~545 - 1 5 5 2
(5)
L.Schmetterer, Research Papers in Statistics (Festschrift for J.Neyman), F.N.
(1964).
David Editor, John Wiley § Sons, London-New York-Sydney 1966,
(6)
J.Wolfowitz, Teor. VeroJatn. Primen 10, 267 - 2 8 1
(1965).
301 - 317.
- 296 -
Two explicit Martin boundaz~ constructions. F. Spitzer.
A.
The boundary for a class of random walks
(joint work with P.Ney, Trans. Amer. Math.
SoT. 1966). Consider random walk on the N-dimensional integers P(x,y)
=
P(o,y-x), mean vector
/J
ZN
with transition function
= ~" x P(o,x) ~ 0 and such that P(o,x) ~ O for
only finitely many x and for some x in each half space. Then the Green function OO
G(x,y)
= ~
Pn(x,y)
n=o
is finite and strictly positive. The function ~(a)
=~
P(o,x)
e
a,x
,
ae
~N'
according to Hennequin has the property that D = E a I subset of R N ,
that grad ~
Finally the sphere ~ - 1
~(a) g 13 is a compact convex
is continuous and non-zero on the boundary
= f x I Xe~N,
Ixl = 1]
b D of D.
is mapped homeomorphically onto
~D
the map SN_1
~
~rad ~(u)
~
B D
= v
~u = ~(v)
Igrad~(u)l It is proved that Z N u SN_ 1 is a compact metric space with the ~etric
I (x,y)
I~-~I
,
x,y ~ z~ ,
I~ - yl
,
x ~ ZN,
Ix - y l
,
x,y
E
y ~ SN_I, SN_ 1
X
, and that in this topology the functions fy(.), defined for each
where ~ 1 + Ixl
y ~ Z N by
[ ~ G(o,x)' fy(x)
x e ZN
=
l
L e
~ (x).y
, x~
SN_ 1
by
- 297 -
are continuous on Z N u ~ _ Corollax-j:
I.
The usual Poisson-Martin-Doob
functions h as
S h(y)
representation for the non-negative harmonic
%(x).y
=
e
p (dx)
~=
probability measure on SN_ i.
SN-I B. The boundary for a class of branching processes
(joint work with H.Kesten, to appear)
Consider a Galton-Watson process with generating function f(s) = I - m + ms, O < m < q , so that the tramsition function P(k,j) = coefficient of s j in ~f(s)~ k =
(~) mJ(1-m) k-j,
k _> j ~> 0 and the Green function oo G(x,j)
[fn(S) ] k,
=
~ n=o
,ith fn =
Pn(k,j)
°
f
°
<
~ , f,
""°
where Pn(k,j) = coefficient of s j in
n times.
It is shown that if N = set of positive integers, T = [ o,1), then
No T
becomes a com-
pact metric space if one introduces the metric I
I
i
9,(x,y) Ix-
yl
,
x,y e T
where ~ is the fractional part of log x / log(~), and that in this topology the functions fj(.), defined for each j e N b y G(x,j),
:~"
e N
fj(x) = xeT "V =-cSO
are continuous on N u T. Corollar~ (proved directly b y H.Dinges): of
(SO k=S
The invariant measures,
i.e. s o i u t i o n s ~
0
-
298
-
have the Poisson representation
/u(j) = J Z j ( x ) v(d=),
j~
~,
T
for different probability measures
~ on T.
Analogous results obtain for all Galton Watson processes with m = ~ k p ~-
k log k Pk ~ ~ "
k
and
- 299 -
STRUCTURE DES LOIS INDEFIN!MENT DIViSIBLES
(2~s
=
G(x))
DANS U N ESPACE VECTORIEL TOPOI.OGiQUE (SEPARE) X. A. T ORTRAT
Introduction et propri~t~s G@n~rales. I - I
D~signons par ~ la topologie de X , par ~ la tribu bor~lienne associ~e ~
(engendr6e par les ouverts de
X
), et par ~ ~ $3 la tribu cylindrique engendr~e par
lem "demi-espaces" ~ x : x * ( x ) < a) , A tout sous-espace ferm~ une m~me classe les
x
Xo
c
non s~par~s par les
X o . E n particulier si Xo =
Y
x : ~
....
est de dimension finie
(points de
Y
base { e i
)
(X,~)
satur@ pour la relation d'~quivalence qui r~unit dams
qu'on d~finit aussi bien par l'ensemble nulles sur
dual de
Xo
X ol
(dual de
Y
est codimension
= xn = 0 , n
x @ ,correspond un espace quotient
, ... x~
, et les valeurs de
) n
des
x~
finie
:
qui sont
lin. ind. x I .... x n
constituent un syst~me de coordonn~es darts
(e±) =
X~
Y = X/X o
sur les Xo-classes Y
, correspondant ~ la
±j }
L a topologie quotient darts
Y
(ou projection sur
compatible avec la structure vectorielle de
Y
Y
: la
de ~ ) est l'unique topologie CV. de
y~
vers
O
~quivaut
x i (y~) -~ O(i = I,... ,n). A chaque dams
Y
Y
est associ~e, dams
X
, qui est dams ~ pour tout
= ye~J~ ~S x
, la tribu Y
~y
das cylindres ~ base bor~lienne
de dimension finie (notons
engendre ~ : ~ = g((~). Si ~6 a
c ~.
). L'alg@bre
d~signe la tribu de Baire (la plus petite
rendant mesurables les fonctions continues, on a en g@n~ral
~c~
Y a ~
:
- 300-
Nous d~signons par ~ la classe des fonctlons continues born@es sur classe de celles de ces fonctions
I - 2
A route mesure
en particulier pour x~(F)
, x~(H)
F
Y
qui sont mesurable ~
d@finie s u r ~
undimensionnel
X
,par ~ '
la
.
au ~ correspondent
des projections
pyF = Fy,
(X o = { x : x*(x) = 0 # ) nous d@signerons par
les lois de telles projections
: ce sont celles des v.a.
x~x)
sur
l'espace mesur~ (X,~,F ou H). Consid@rons maintenant lois ind@finiment (q)
p a S
les lois de probabilit~ ~ d@finies s u r ~
divisibles
~
ll est @vident que doric sur ~ . projection
tout 3n
A E ~
de ~
(Uy, i
ou ~ )
est unique s u r ~
voisinage ~Y
en indice
=
de
Y
on a la d@composition
de
Y ~
,
P . LEVY de la
:
0
darts Y
est normale,
et
~ y = e(Fy) Fy, i
de
Fy
est " du type de ~
Y - Uy, i
) est born~e et que (pour une suite de
i-*~lim e(Fv,i)xl~
(~ Y)
telle que
car unique en projection sur chaque
(E J0 (Y)) ; c'est dire que la restriction
(3) xi
~n(SUr~
doric tout A ~
unique ~ une translation pros, od ~ y Poisson "
n
De plus dans chacun de ces py
et la classe ~ des
( au sens strict qui suit ) :
il existe pour tout
n (A) = ~ A (+)
ou 2 ,
d§signant la
,
Uy, i4 O) on a:
classe ~ ( Y )
translation
oper@e sur
e(Fy,i),
et
e( v )
d@-
signant pour route mesure born@e ~ la loi (3')
Darts
e(~ ) = e- ~
(3)
~
~n ' ~ o= ~ (0) El-
on peut supprimer la
translation
translation fixe (ind~pendante D@finition
i
si
F~
q
en
O.
est born@e,
) si on centre ~ ( et si
ou mettre une Fy
est born~e).
:
(+) Nous @crivons la convolution produit,
de
masse
des lois (ici associative
sans signe interm~diaire.
et commutasous forme de
-
Appelons loi impropre une lei dont les
30~
x~(
-
) sont impropres. Lorsque
X
est locale-
merit convexe, une telle loi est une constamte ( son support est r~duit ~ u m point de X
).
I - 3
U n premier probl~me regoit ume r~ponse tout ~ fair g&n~rale (cf. I - 5) :
P o u r toute p tendue a(q), on a ~ = p ~ et ~ y
, p et~ sont tendues et leurs projections sont
respectivement
~y
de (2) (Y
variant dams ~ o n
patible de ~ T
, par example en centrsnt les
Y
peut choisir um syst~me com-
, alors le syst~me des ~ y
aussi). Cette d§composition est unique ~ une loi
i~propre pros, p e t
l'est
v sont caract&ri-
s&es par : a)
~ est gaussienne
b) ~ a dea
~y(Y~
~)
: les
x ~ ( 9 ) (doric les ~y, Y ~ ~ )
du type de Poisson
sont normales ,
(a (3)).
Mais l'important est (deuxi~me probl@me) de montrer que ~ = e(F) F ayant les
Fy
de (3) projections.
tiom ~+)p = p V
mesure
~ (X, ~ )
, ayan
Darts tout groupe ab&lien m~trisable une factorisa-
, avec ~ = e(F) (d~fini comme en (3)) est assur~e, mais avec une caract6-
risation de 9 trop faible qui n'assure pas
a) ( on sait seulement que ~ a ~ et n'a pas
de facteur "propre" du type (3'). Dams la partie II nous croyons utile de reprendre l'~tude de cette d~composition dans le cas od m~trisable),
suivant les id6es de ~8~ ( cas od
nombrable de voisingages. Nous 6tudions aussi, d~finition de
e(F)
d@nombrables
(of. ~#~ , et ~ 0 ~
on des mesures s~parable,
F
0
ou
est um groupe ab~lien (g6n~ralement X
localement compact a une base d~-
l'unicit~ de
lorsque le groupe ab~lien
l'extension ~ ce cas du th~or~me de
X
I
X
F
(cf. le lemme 3), la
n'est pas m~trisable et notons
de P.
LEVY relatif aux convolutions
). Nous n'abordons pas le probl@me de la caract~risati-
susceptibles de d~finir ~ = e(F) (darts le cas od
il est n~cessaire et suffisant que JIIxll 2 d F ~
Darts la partie III, supposamt que
X
X
est hilbertien
, cf. [q#~ ).
est s~parable et admet une base
(d~nombrable,
avec des coordonn~es continues) nous montrons que route loi ~ tendue de ~ ( X , ~ )
satis-
fait ~ :
(+)
Avec l'hypoth~se restrictive
les A n
(si
X n'est pas ~ base d6nombrable de vQisinages)
que
de (q) sont tendues: ne sachant pas en g@n~ral si tout facteur d'une loi tendue
est tendu,nous devrons ne prendre en consi4~ration que des facteurs(et cofact.)tendus.
- 302 -
(~) X
~
= ~ e(F),
x~(~)
peut n'etre pas m~trisable,
e(F)
normales,
F
unique.
et il faut utiliser l'extension de la d@finition de
~ ce cas.
Dams les conditions de la partie IV, la m~me r~ponse r&sulte de ce que H tendue de
(x,~) (+)
satisfait ~ # = lim e(n~/n) mais
X
doit ~tre suppos~ m~trisable pour utili-
ser le r~sultat g~n~ral de la partie IX. Dams la pattie V , une autre m~thode
(part-
iellemen~ u~ilis~e dams ~ q ~
), indirecte, permet de lever l'hypcth~se de la base,
dans III, mais en supposant
X ~ faiblement s~parable,
trisable,
I - $
(X, ~ )
polonais s~parable, m~-
un espace vectoriel topologique (s@par~).
Rappelons que la loi p s u r ~
est dire tendue (ou ~ -tendue pour rappeler le lieu de
avec ~ ) si ~ des compacts ~ B = sup KcB
Lorsque ~ (5)
X
complet.
Soit
alors ~
et
~B(B~
,
Ke K
(donc de ~ )
compact ) et nous ferons toujours cette r~gularisation.
est seulement d~finie s u r ~ A ~ ~
et
A n KE
avec p K~ ~ ¢ . ~ r~gularis~e satisfait
= ~ ~pA
, nous la dirons ( ~ )
tendue s i R K ~
tel que z
< ~ .
Alors suivant um lemme de Prohorov route fonction p a d d i t i v e ~ O, d~finie s ~ r ~ e t faisant ~ (5) est ~ -additive (++) donc prolongeable ~ ( e t la propri~t~ fondamentale suivante
~ (5) s u r ~ g ) .
satis-
Rappelons
:
Lemme q : Csazonov):Lorsque
X
est localement convexe (s~par~) toute loi ~ tendue, d~finie sur
, est prolongeable de fagon unique ~ ~ , en une loi r~guli~re, bien il suffit que les
(+)
Les
~n =
x ~ a X ~ s@parent les points de
X
/n de (q) sont automatiquement tendues,
(++) Cette fonction ~ doit ~tre r~guli4r~,
et tendue. Aussi
.
sous ces conditions.
c'est ~ dire approch~e par les ferm~s de
, mais cette condition est v6rifi~e par routes les ~ q u i sont ~ -additives sur chaque
~y(Y ¢ ~),
ou mesures faibles au sens de certains auteurs.
- 303 -
D@finition : py
et py,y
(Y'DY)
famille {Py} Y'oY
~
d~signant les projections de
de mesures d@finies sur les J~y
X
sur
(Y g ~ )
Y
et
Y'
sur
Y
, une
est dire compatible si
Pyy,~y, = py.
Lemme 2 : Si pour tout
Y¢~
, etp
tendue d@finie s u r ~
(6)
P~
les 9 y
@tant lois s~m@tris@es
= P~
, on a :
~ (gy :~y
q~
)
et les syst@mes [ g Y }
cun compatible, alors il existe des lois 9 et ~ tendues (uniques, e ( X ~ ) ) H =P~
, p y 9 = py , pyV
=gy
, {Wy}
cha-
telles que •
.
La conclusion est la meme si au lieu de supposer les 9 y sym@tris@es, on suppose 9y
= py9
(?tendue
&(X,~)).
Sip
est donn@e tendue (r~guli@re) surJ5 , et
X
lo-
calement convexe, la conclusion s'6tend ~ ~5 . D@monstration : Soit
K
py~
un compact E (5) (a fix@). Sa projection
>1-8.
lation
-x
Soit~= ~'~'', op6r~e sur
pA =
y
A
et d@signant par
K'
~y.
Sy
~
A
d@signant la trans-
sup
x
~ " (A- x )
A g
Vy
la difference vectorielle @
py(K'y)>~
doric il existe ~ tendue avec 9y = PY9 =
-x attach@
, vule
K @K
, on a
"l-2a , lemme de Prohorov. De,plus on a , avec
KeK' I-~.<
et
x g p ' ' ( A _ x ) > q- a ; appliquant ceci darts Y
~y(Ky,_x) >~1- E
K''
est un compact, donc de ~ y
)
p'(dx) p"(A_x).<
entraine l'existence d'un ~y =
l'~galit@ (l'indice
~
9 y ~y Ky .< £ py(dx) V y ( ~ , _ x ) + 2£ .~ 9(K'~) + 26 " K 'y
.
- 304 -
Toutes les affirmations du lemme s'en d@duisent. Corollair~ I z Si deux lois p , ~' ~(p),
tendues (sur~g)
ont leurs p y ,
x*(p') tramslat@es l'une de l'autre, ~=
clusion s'@tend ~ ~8 (avec
a = constante) si
~'T~Y~ ~),
~'. a, X
a
ou simplement leurs
@rant une loi impropre.La con-
est localement convexe ( et ~
,~'
r@guli@res ). Corollaire 2 : Toute loi p tendue, s u r ~
, ~
x~( 9 ) normales peut @tre centr@e ( au sens faible, cf.
[1], [7~ ) et appartient ~ ~(X,~ )(avec si
X
1/n tendue). De m~me sur m (avec ~ r@guli@re) ) sont U.T. ( suivant les K E!
est localement convexe. De plus l e s 9 t ( O < t ~ 1
et 2~ ), donc le semi-groupe t
est d~finl (de fa@on unique) et continue pour la CV-
en loi. (+) Remarques z 1)
I1 faut, si
X
n'est pas localement convexe, entendre "peut ~tre centr6e" au
sens de : par convolution avec une loi impropre a ( les tes, le support de E{x~(~c)
~
a
a
jouent le r~le de constan-
a ses points non s@par@s par la tribu ~ )
= O. Dams le cas localement convexe, ce point
: 9 = ~c
a,
(~ centrable ) a 6t@ d@-
montr6 par AHMAD (cf. [1] ) , au moyen de propri@t@s plus g@n@rales (moins simples, @rude de la d@finition d'une moyenne faible). 2)
Que ~ G ~ , es~ une propri@t@ qui n'est pas exacte pour les groupes non vectoriels,
d'apr!@s l'exemple donn@ par URBANIK : le tore compact de dimension 2, cf. [13~ ( remplasaht les
x ~ par les homomorphismes de
X
sur la c£rconf@rence ).
Corollaire ~ • Si ~ tendue £(X,~S), est telle que p y
~S(Y) (pour tout
Ye~),
la loi sym@tris~e
U s = ~ l -'-~ S(x, ~ ) (nous ne savons pas montrer en g@n@ral,sauf ~gaussienne oustable, cf. corollaires ~,
(+) Lorsqu'il s'agit de lois sur ~ (
CV. d e s
/.~f = ~fb d/.J
non prolongeables ~ ~ ), la CV. en loi signifie la
pour chaque f a ~'
(,au l i e u
de
~).
- 305 -
et 5, que H e l l e et continu.
m~me s U ) De m@me que ci-dessus,
le semi-groupe
~ t S
est bien d@fini
(+)
Corollaire 4 : Toute loi ~ tendue d@finie sur ~
, g projections
culier tou~e loi ~endue de ~ ( X , ~ ) pr@s) en p = p ~
, odp,~
a)
les
x~(9)
b)
les ~ y ( Y g ~ )
p y ~ S (Y)
se d@compose de fa$on unique ( d u n e
), en partiloi impzopre
tendues sont telles que :
sont normales, ~ ~ ~
et peut ~tre centr@e,
sont du type de Poisson.
Cette d6composition se conserve par projection sur tout prolonge g ~
(tout Y E ~
lorsque
X
Y =X/X o (cf. le n°1), et se
est localement convexe.
Les d@monstration de ces corollaires sont imm6diates, Corollaire ~ : Toute loi ~ tendue d@finie sur ~ et ~ ~ y
stables
(Y ~ ~
)
est ind@finiment
divisible
et stable ( le prolongement g ~ de cet enonc§ vaut @videmment dams le cas localement convexe). D@monstration P o u r chaque de lois
y
: c' ~ 0
Y, et c,
donn~s, il exi~te d ~
v.a. ind~pendantes
~,
Z~'
,
satisfaisant
c
+c,
c'' ne d@pend pas de cette loi (vu
c,, (Y) Y
car
Y'oY
c ~ + c '~ = c'' % ) .
÷a(Y) ~
c''(Y') = c''(Y), doric non plus l ' i n d i c e %
Les lois de
c ~,
c' ~
, c'' ~ '
comme projections des lois de c Z, c' Z, c'' Z, Z de loi ~ , on a impropre,
~,
d'apr@s le lemme 2. Ainsi p e s t
c Z + c'Z = c''Z'' + a , e t a 2 ,
car (avec
est ~l/n
c.a.d, qu'on peut faire de c, c', c'').
(en fair
@tant compatibles,
a(Y) = py a , a
une loi stable, en ce sens que Zl, .... ,Zn ind@pendantes,
de lois ~ ) ZI+Z2+ .... +Z n = b n Z + an , a n impropre, ( Z -_ an/n )/b n
de
et, comme Z ,
entraine que la loi de
b n = n 1/~ ).Pour ~ ~ ~ , v peut ~tre "centr@e"
a = 0 (apr~s translations
~-
~/(c''-c-c')
ind@pem~amtes
- 306 -
Notons qu'on montre comme dams
Rn
que les lois stables (tendues) constituent routes
les lois limites norm@es ~ termes bien d@fini et continu,
ind@pendants
car apr@s centrage
d @ I , V t est
de m~me loi. Pour
on a
t t~O
tout voisinage ~ , @quilibr~ et absorbam~, geables ~ ~ , il faut prendre
~a~,
',(cf. la note de la page I - 5
Th@o~@me "i :
Si les
Fy
[Fy~
de O. (Si les lois s u r ~
de qui correspond ~ la CV.
ne sont pas prolon-
ptf
--. f(O) t~O
, sur
).
du corollaire 4 ci-dessus
(avec ~ y = e(Fy)) sont de variations
totales
telles que
sup [Fz] = M < ® alors il existe
F
= p e(F)
,
cette factorisation am ~ ( ~ )
(X, ~ ), telle que
, mesure born6e tendue sur les
x*(p)
sont normales
;
est unique ~ une loi impropre pr@s, et
engendr@ par l'anneau
~ =y2~4
, ~y
F
est unique sur le ~ - s h o e -
@rant la r@striction de ~ y
-I I o. 1 Notons
que H a ~ ( X , ~ ) ,
, si
X
est localement
D@monstration Compl@tant
si on ne l'a pas suppos@ au d@part,
chaque
Fy Fy
,dams
= ~ c ) et les
Fy
3a et les
.
3~ eM
, en ajoutant en
K~'
0
la masse
R des
aT
M-[Fy]
, on voit fa-
tels que les ~ y = e(Fy) ay
du lemme 2) : Noter que 9
@rant centr@e,
(soit
born@es, une translation intervient dans la d~finition de ~ y
c'est (au sens de (3')) et
Y
deviennent compatibles.
satisfont ~ (5) (avec
K~'
donn~e (reguli@re et tendue ) s u r ~
:
cilement que les
les
convexe et ~
et que l'~nonc@ vaut sur
Vy
= e(Fy)ay
d'o~ l'existence
.Donc de
les
FEpyF
aT Fy,
et
(Fy)ay et ~
:
satisfont ~ (5) avec les
9c a e(F)
La factorisation ~ ~ obtenue est la seule (~ une loi impropre pr@s) telle que ~ soit "de Poisson"
et~ de ~ sans facteur propre
e(G)
car s i ~ = p' v'
6tait une autre. F'' = F - F'serait n@cessairement ~ 0
, avec ~' = e
en projection sur tout
en
Y-O(Y~),
- 307 -
donc nulle sur les tribus JSy : sinon on aurait : @y = 9'
eCF~') " ~
::~)~ = p y
e(F]~') ::~ p' = g e ( F ' ' ) ,
aurait des facteurs propres "de Poisson".
Remarqu~
:
Nous n'affirmons pas en g@n~ral l!unicit@ de
F
sur ~ O
I X-O~ = ~ .
Lemme ~ : S'il existe dams le dual Yn
X ~ de X
s~parant les points de
X
(espace vectoriel topologique s@par@ )
,~(~)
=~:
bien d@finie par ses projections sur les
route mesure s u r ~
, nulle en
une s~ite
0
, est
Y-O (Ya ~ ). C'est ~ dire que & est f~m~lle
d'unicit6 pour les mesures H tendues sur a~ (ou ~8 dams le cas localement convexe). D~monstration : sst engendr~s par dfini par
~ , .... ,
U [~y Y
,et si
UI~ ' ,m
non tous nuls J ,
- o~
, avec ~
d~signe
x : xiCx)
on a
- 0 =
= ~1
{o} ; mais si
Y
est
x) = o, F~,m, J' car la suite
--Y~ @tant
m=
s@parante, {0 1 = t o o 1
(X-E~,m).
Ainsi ~ , m ~ ~
~
~ - 0 ~(&)
-
II
•
-
Th@or@me ~ n ~ r a l de d@composition 9 = H v II - 1
~ ~ c ~ (~)
dans un ~roupe ab~lien m@trisable
Nous donnons certaines propositions pour
X
.
non m@trisable, cf. en particu-
lier le lemme 7 et n°II-3. D@finitions : 1)
~ a ~
constante
(ou ~ (X)), classe des lois ind@finiment divisioles, si pour tout aha X n =~n
et une loi ~ n ~n
'
tendue telles que ~ n tendue ,
n 3 fLUe
- 308 -
motant
mutiplicativement,
notamt
a
2)
~ a ~
sans signe interm~diaire,
route loi d~g~n@r@e
, de support
est facteur propre de p ( ~ a
3) p ~ ~ o m = m
4)
2
a .
~ )
est dit cofacteur de ~ et on ~ c r i t ~ 4 ~
si on a ~ = ~
, avec ~ ~ (
et~,vtendues);
.
si ~ ~ ~ et s i p n'a pas de facteur propre idempotent m. En fair tout facteur
dep est facteur propre pui~que ~ = m ~
Une f~mille I ~
si 3 des
la convolution des mesures, et
K~
(£~0
~
--~ ~ = m ~
.
de lois tendues est @quitendue (ou uniformement t e n d u e :
arbitraire ) tels q u e ~
K£>l-a
ment oe mesures bern~es, on ajoute la condition ~
, pour t o u t ~
~ ~ M, pour t o u t ~
faible, des lois ), m~trisaole (+) si
r~ciproquement,
lorsque
X
est m~trisable,
X
)
. S'il s'agit seule.
Rappelons qu'une famille ~quitendue de mesures est relativement compacte convergence,
U.T.
( pour la
l'est (cf. ~11~ ) et que
tout suite convergente on route f~m~lle
relativement compacte de lois tendues est U.T. (++) Ce qui suit ( apr~s le lemme 4) repose essentiellement sur les proposition correspondantes de ~ 8 ~ ( et des r~sultats de ElO~ , ~11]), nous croyons pr~f6rable de redonner les d~monstrations,
ais~ment ~tendues au cas envisag@ ici od
merit compact, ni "s@parable"
X
n'est pas mi locale-
( = ~ base d~nombrable de voisinages),
et pour certaines
,
au cas m~trisable. Lemme 4 : Dams tout groupe topologique ab§lien, les lois ~ de ~ sent de la forme m = mH
est la loi de
satisfaisant ~ ~ x = p ,
Haar od
Si X n'&tait pas ab~lien,
(~tendue ~ ( X , ~ ) ) ~ = X/H, ~ y
sur le sous-groupe compact
est la projection de ~ sur
il faudrait supposer
m × ~y,
H
Y
H
et~ya
sous-groupe normal de
X
od des
x
~o(Y)pour que
ce l~mme reste valable. D§monstration
(+)
:
I1 suffit que les
K~ sous-tendamt la famille soient m~trisables(si X me l'est pas).
(++)Th&or~me du ~ VAR~ARAJAN, cit& par PROHOROV (4@me Symposium deBerkeley)cf,
aussi[2].
- 3O9 -
I~_¢f. [11] . L'@galit@ p m = p se r@duit en effet (7)
p m A = ~ py(dy) m ( ~ ) yAy=
AnyH
quelque soit le point produit
H × Y. (~)
H). Que p y
y
, ~
= section de
A
d@finit par
,
choisi dams ume H-classe pour repr@senter
exprime bien, @galement, le produit
appartienne ~ ~o(Y)
X
m X~y( m @tarot alors restreinte
est ~vident, et ceci ram@he l'@tude de
celle de ~o(X). Notons cependamt que dams E~] les auteurs ont relev@ ~y sous la forme de p' a To(X) p = mx/~
comme l'espace
~(X)
dams
X
,
:
= mxp~
(/~ = ~
= m/~'
ce qui permet d'@crire la f.c. de p ~ ~, dams
X
)
,
(en facteur
s'introduit la f.c. de m ).
Lemme ~ : Dams tout groupe topologique ab@lien, les lois tendues de ~ forment um semi-groupe fern@. D@monstration : Consid@rons ume loi l~m~te p (suivamt um certain filtre) :
8i p ~ = ~ n (pour
n a~n
, il existe des
n fix@) U.T.
, et les
b~n
a~lm bn~n
rendamt les
~a b ~n' doric les ~
n
n
appartiennent alors ~ un meme compact (comme
cofacteurs de facteurs U.T. des p~U.T.). On en d@duit une loi ~n =~nn
l~mite de cn
~n
et ume constante
cn telles que :
doric H ~ ~"
Lemme 6 : Darts
X
groupe topologique, pour toute loi ~ tendue, l'ensemble des facteurs ~ gaucae
(tendus ainsi que leurs cofacteurs) de ~ est ferm6 pour la CV. faible U.T. De m~me, si X est ab~lien, l'ensemble des facteurs propres de ~ tendue ~ ~ , U.T.. En particulier (X
est fern@ pour la CV.
~tamt ab~lien), pour route loi p ~endue de ~ o '
on peut dams
-
(1') supposer ~ n ~ H
e
, et pour toute p g 7 ,
@tamt le sous-groupe
D@monstration
310
-
supposer ~ n - ~
compact maximum invariant p
mH
et Zn
H-invariante,
, d@fini au lemme @.
:
Soit p = ~ v ~
et ~ limite U.T. d e ~
, les ~
sont alors aussi U.T. et on en d@-
duit (avec 9 limite de vg suivant un filtre assez fin ) p = ~ lemme 5, si ~ ,
va g ~
En particulier
soit p
. De m~me,
suivant le
,9 et ~ a ~ .
tendue et de Y
. I1 existe des
bn
tels que les Zn' = ~n bn
(avec Z n E (1')) soient U.T.,
et pour route limiteZ de tels Zn' (soit ~'n~ - ~ ~ ), A ,n~ k divisep (pouz n a & k) ; les ~ k divisent d o n c p et sont U.T. apr@s translation doric (cf. ~11~)
~
a H, et (Z a-l) k
k-~oo
H = H' = e ,~ = e , e t Z n ' - ~ e Si non, on a (avec avec
~n'-~e m×
II
-
Soit
2
Y
~n' = ~ n est
divisant H
; modifiant
Y = X/H)
, dams
mH'
~y
; doric p =
les
= ~ ' nn
Yn
am
doric H ' c H .
et
'
Yn
'
~ n - ~ mH(H' = H ).
Lemme 7 : X
groupe topologique ab~lien et p tendue ap@riodique.
donn@ de
e
divise~,
ait un cofacteur tendu et soit de variation totale
bien s i p
~Jo
, il existe une mesure born@e tendue
D@monstration a)
(5)
est ap@riodique
de (2) , c'est bien ~n---e.
(m x ~'n )nyn = ~
H-invariante
Sip
' et si les
U
voisinage
, darts X-U , telle que
ouvert
e(F)
[F~ = f maximum. Aussi
sont des facteurs propres.
:
les mesures
e(F)
sont born@es,
e(F)
F
Pour
F
(born@es,
divise ~ ,
tendues) telles que
F(u) = 0 ,
car s'il existait une suite
(on peut supposer) fn~n, posant
Pm = T
Fn
Fn
avec
fnT®
,donc
, les ~n = e(F~) seraient tendues et telles
n q u e a nn d i v i s e ~ ( v u F n = n F~ + (fn - n) F n' ) . Le raisonnement du lemme 6 montre qu'une suite de A n convergerait apr@s translations, vers e , doric aussi les
-
e(F~ + F'~ n ) : ~ns ns b)
-
des ~ n ; cela est impossible puisque
(X-U)~ e-2(F~ + F~-)(X-U)> 2 e-2 devrait tendre vers sup IF] M = F~(5) , et
Soit
U.T.
sym~tris~es
311
, suivant des
K~
F' = F + F- ; les
F
sont U.T.
cette suite fournit La d6monstration
sym~tris@es
e(F)
des
sont
, d'ou e2M
ainsi les
e(F')
0 .
, et si e(F)
(K c
X
K),
est une suite e [ F n ] ~ M , route mesure
Fn
F
adh@rente
facteur d e p , avec [F~ = ~.
est la meme (cf.
[8~ ) pour des
e(F)
facteurs propres de ~ e
~o"
Th@or@me II : Dans
X
groupe topologlque
ab@lien m@trisable,
route loi ~ temdue de 70, admet, dans
o' une d@composition
(8)
ju
: .p,~
telle que 1°) ~ ~ ~ o
: ~ est ap@riodique
(9)
= lira e(Fi) x i
les
V
G i = F i - Fi_ 1
n@e port@e par
~ e
~o
, avec
(9) : ~i/F,
@tant des mesures
0orn@es "port@es" par Ui_q-U i
X-U i) , r6pondant au lemme 7 pour
= e (Fi_ 1 ) ~ i-I 2° )
eta6
et 9i-1
Fi
bor-
, avec
;
et n'admet aucun facteur propre du type
Nous &irons que 9 est une composante normale de p D@momstration
Ui
(donc
e(F).
, et que la loi ~
est normale.
:
Le lemme 7, avec
U1
, nous fournit
~U = s ( F 1 ) ~ 1 On recommence avec ~oI
' 9q ~ ~o et
F1 = G1
, m~Timum en variation,
avec
"
U 2 , d'ou ~1 = e ( G 2 ) 9 2 '
G2 ~tant port@e par
UI-U 2
(si-
512
-
non la restriction non nulle de
G2
~
X-U 1
-
permettrait "d'agrandir" le facteur
e(F1)
et ainsi de suite : ?i-I = e(Gi) ~ i La convolution xn
e(F n)
U.T. e t e ~o' donc s i ~
3 0)
~
converge lorsque
e(G,).., e(Gn) = e(Fn)
convenables, les
P
"
= 9 V
n-~,
apr~s translation
@rant facteurs de P tendue les cofacteurs
~l~n
sont
est une de leurs lois limites on a | , peso
' ~ £ (6).
est facteur propre de tout 9i
~a le lemme 3 et :
? i = e(Gi+1)'''e(Gn) Xn Xn I 9 n
'
doric e(~) s'il existe ~ortiori
<~
~
e(~)4pi
,
pour tout
~ ~ e , ~i restriction de
e(~i) < ~ i ' donc
lativement ~ ~ i-I
~ ~ X-U i
e(G i) e(~i) 4 ~
i
; est ~ e pour
i
assez grand, et
i-~ 4 p , et [ Gi~ ne serait pas m~yimum re-
"
Remarque : On rencontre des difficult@s pour lever l'hypoth~se que est de savoir c~oisir les alors
X
est m@trisable : l'essentiel
(Ui,F i) r@pondant au lemae 7 tels que
UjC Ui ~
Fj ~ F i ,
F = ~
F i d@finirait F (cf. les lemmes 8 et 9 ci-dessous). Darts le cas de i la pattie III nous tournons cette difficult@. II - 3 X
D@finition de la classe de P o i s s o n ~ ( X )
dans un groupe topologique ab@lien
.
Darts ce qui suit nous @tudions la d@finition la plus g@n@rale de
e(F)
dams le cas
non m6trisable. Son int@r~t tient en partie, nous semble-t-il, ~ la proposition suivante dont la d@monstration est une extension facile ( que nous ne donnons pas ici) de celle de [10J (Th. 4, cf. aussi ~4] ) : Th@or@me Soit
:
(X,~5)
un groupe topologique muni de sa tribu bor@lienne et 9i(ia I)
une fa-
-
313
-
mille de mesures born@es tendues d@finies sur ~ ) , f~m~lle filtrante en ce sens qu'~ tout couple
i,i'
de
I
correspond
facteurs ~ gauche (+) de
au moins un
~j (d'od l'ordre sur
d@signe le filtre des sections ~ droite de
je I
tel que
I : i<j~-~V~
Vi, vi,
, soient des
= ~i Vii ' i ~ j ).
I. Alors :
OU
sup Vi x(K) - ~ 0
, pour tout compact
K
( de m~me pour les ~ i ~
' i ~ j , chaque
i
) |
X
OU
il existe des constantes
ai
telles que les ~i ai soient U.T.
Lemme 8 : Soit ~ , ~ g )
un espace de mesure, Fi(ie I)
un ensemble de mesures d~finies s u r ~ ,
filtrant croissant ( au sens ci-dessus). La relation F = sup F i B i d6finit une mesure sur
(X,~)
(limite I , suivant ~-des
Fi).
D6monstration z a)
F
est ~ O, additive car
i, i', j
sont tels que
F(B+B') = sup i
Fi B > F
B-~
(F i B+F i B')~ F B+F B'
, Fi, B'> F B'-g
et F ~ > ~ i
F(B+B')>~ ~j(B+B') = Fj(B) + Fj~B')~> F i B + F i, B ' > F B
, mais si
" Fi'
, on a :
+ FB' - 2~ .
N
b)
mais FB = sup =
co ~-I
oo Fi i Bn'< ~
sup i
Fi Bn
I
FBn
Lemme ~ s F
~tant d6finie suivant le lem~e 8 , avec des
topologique ab61ien ) on suppose que les
Fi
e(Fi)a i
sont U.T. (2@me alternative du th@or@me ci-dessus) (+7 avec cofacteurs tendus
born~es, sur
(X,~5)
(X groupe
( apr@s translations convenables et que leurs lois limites sont
a i)
-
314
-
[email protected] ces lois limites sent @gales ~ une translation pr@s. (Dams le cas d'une convolution d@nombrable of. B.M. KLOSS, r~f. dans ~ 0 ]
, et [4~ , [11~ ).
D~monstration : D~signons par
Fij = inf (Fi, Fj)
example par sa densit@ p.r.
m = F i + Fj
inf ( ~--m-- , On a
Fij ~ Fj
la borne sup@rieure des mesures ~ Fi,F j ( d@finie par , qui est
) .
( et sym@triquame~t, suivant le f i l t r e ~ - o u tout autre plus fin). Fj
@rant born@e (done les
Fij
) on voit facilement que
e(Fij) ~ e(Fj)
( et sym@~rique-
ment) dams les memes conditions. • Soit alors .
e(Fi)a i
Y' ~
pour deux lois ~' , v'' e(Fi), '
e(Fj) d i v i ~
cste~et
~'
et
e(F j) aj
~'' ~ ~''
,
adh@rentes ~ la famille U.T. donn@e. Puisque v'
( limite suivant
et sym~triqueme~t, d'od
~ 6 ~ =
~'
divise
~'), puis (limite suivant ~'') ~ ''
(raisonnement classique)
constantes, puisque
e(Fij)
~' , ~''
~ ' = G ~'', V'' = ~' ~'
divise ~ ° ~
sent supposes aperiodiques.
Cela s'applique (sans cette restriction) ~ tout espace vectoriel tcpologique puisqu'il ne poss@de pas d'el@ment compact, done de loi p@riodique (cf. [10] ). Lemme a)
10
:
La loi limite ~ = e(F)
d@finie ~ une translation pr@s, darts les conditions du
lemme 9, sur le groupe topologique ab@lien obtenue en prenant la famille 7) aux
U~ = X-U i
( {Uil
F~
X. et sa tribu bor~lienne ~ ,
@gale celle
des restrictions (nec@ssairement born~es v u l e
base de voisinages de
e
) et ne d@pend done ni des
lemme Fi~ F,
ni de la base ~Uil b)
Pour route famille filtrante de
suivant
a)
, ont
F~
born@es ou non, ~ F
,les
e(F~)
( apr@s translations convenables) pour seule loi limite
d6finies e(F) ,
une translation pr@s. Lorsque
X
est m@trisaole (I d~nombrable), ou lorsque il existe des
sous-tendant V , m@trisables, on peut choisir les
a i'
de sorte que
K~
(~ i t O) i e(F~) ~ - - ~ e(F).
-
D~monstration a)
Soit
315
-
:
Fij
la restriction de
Fij ~ F'j @
Fi
~
Ujc
e(Fij) ~ e ( F ' j )
; on a
,
et e(Fij) F'S
divise
est donc born@e et V = e(F's)~S
Puisque
e(Fi)-~
F'j ~ F
e(F~)
divise
auxmSmes
( car sinom il existerait une l o i ~
xi
tels que
~
~j(i) x~-~ e , d'od
F'~ F. Les restrictions U ci ) et ~ F'
donc les
translations, doric e(F')
telle que
e(Fij(i))--~
e , & v e c v~ = v
, ce qui
F i, e(F~)
de
F'
non bor~ees) et les
Fij
e(F~(i)) xi---v , et la limite
e(F~) a~ U ic
aux
U.T.
, ~gale V .
sont 4 F i (restrictions de
sont (comme facteurs de
peut etre d~finie suivant
stration identique g la pr~c~aente, mais od les
II - 4
avec ~ i j = e(F~ - Fij)-
j(i)
e(Fij)-*e(Fi) , il existe
unique (~ une tramslation pros) d@finie par les Soit
,
= e(FiS)~ j
''V est ap~riodique ").
On a donc des
b)
e(F).
donc
= e(FiJ ) G i j ~ j
et que
domc ~ij(i) ' x'i --~e contredirait
e(F) = ~ ~
les restrictions de
Fi F~
e(F)) U.T. apr~s
a). On peut reprendre une d~monde
g
F
a ) deviennent les
F!~l F(F~
U~, qui ~ F i.
Remarques concernamt la composamte "normale" d~ th§or~me I!.
La conservation de la d@composition (8) du th@or@me II, par projection sur la groupe quotient I° )
Y = X/X o
est @vidente si l'on sait montrer :
que toute loi ~ d e s (ou 7 o) est limite de lois
composante "normale" ~ de ~ ~ o ' U c = X-U 2° )
(10)
( U voisinage de
on a
Gn u C - * 0
e(Gn)X~ . Alors s'il s'agit de la
(suivant le m@me filtre) pour tout
e ) , suivant le lemme 11 ci-dessous.
que inversement route loi ~ de ~ ( o u 7 o) , telle que
9 = lim e(G m) x~ ,
G n U c--~ 0
,
-
n'a pas de facteur propre du type
316
-
e(G) .
Ces points sont assur@s darts les conditions de [8~ . 2 ° ) l'est ~galement dams tout espace vectoriel, finie, H y
= ~y ~y
car (10) se conserve par projection,
doric pour tout
est la d~composition clasique de P. LEVY, ~ y
saurait admettre le facteur le(G)~
= e(~)
Y
.(+)
Y
de dimension
est normale et ne
1 ° ) le s e r a ~ a n s les conditions des
parties IV et V. Lemme 11
:
Soit
groupe topologique ab@lien. S i 2 ~
X
~, ~ = l i m e
saivant un certain filtre) et si ~ n'a pas de facteur Gn uC--*O D~monstrat~qn
,
, alors
I Gn
par
0 e t ~ 1 , et U.T.
, de variations totales ~ q >
sont lois sym~tris6es et facteurs des divise ~ ,
e(G)
(limite suppos@e U.T.
tout U .
Sinon nous pourrions extraire de ces Uc
(G n) x n
ces
( au sens
e(Gn) x n
G'% G)
des
puisque les
G~
port~es
e(G~ + G' n )
U.T.. D'o~ une G llmite telle que e(G)
car les cofacteurs sont aussi U.T.
Ce facteur
e(G)
de ~ est d~ailleurs facteur propre de 9 , v u l e
lemme 6 ; le lemme
vaut donc aussi bien sous l'hpoth~se plus faible que ~ n'a pas de facteur propre
Cas
III
e(G).
-
od (X I ~ ~ est un espace vectoriel topoloHique (s~par~) poss~dant une base .
III-
(11) Posons
1
Soit ~ e n } cette base d~nombrable, Par d~finition, x = lim n~oo X'n = Pn x , Pn
xn ,
n xn = Z 1
d~signa.nt
x(i) e i
la projection
(+) le m~me raisonnement vaut s'il exis~e un p o u r le groupe m~trisable
ab&lien donn~ X
• sur
X
on a, de f a ~ o n u m i q u e
l'espace
En
engendr~
par
localemen~ compact ~ oase d~nombrable
-
el,..,e n ~ on a
; de m~me nous d@signons par
x~ = P m n X m
et u n ensemble
dit syst~me compatible de x (i) (x) e X ~ ,) Remar~ea 1)
Pn
-
Pm n
la projection de
Em
sur
E n (m> n)
{ x m ~ E n , n = ~,'''I v@rifiant ces relations est
. Nous supposons de plus les
x (i)
continues ( alors
sont applications lin@ares continues de
X
sur
En .
:
La continuit@ des
vectoriel Y
et les
xm
317
En
est condition n@c@ssaire et suffisamte pour que l'espace
(avec sa topologie unique,
au sens de x :
x (i)
I-I
....
sur le corps r@el ) soit l'espace quotient
et de tout ce texte (alors
=
= 0
Y * = X o±
, X o = {en+ 1,..~ =
). Cette condition est automatiquement v~rifie lorsque
X
est espace de Banach. 0n sait en effet (cf. [6], pp. 87-68) que la base peut @tre tendue monotone c.g.d,
telle que
II Xn~[/[Ix~l ; alors
I[pnll
= 1 (comme projecteur darts
X). 2)
Ce th@orgme de la base monotone implique (lorsque ume oase existe ) la propri@t§
commue que ~ = ~
, car la boule llxll ~ 1
sont dams ~ . Em g@n@ral (lorsque = ~a
~
, l'alg~bre
U~ n , n
a ume base ms_is n'est pas de Banach), on a
En
(~y
pour
Y = En
n
@tant la tribu des cylindres ~ base bor@lienne
avec les notations de
p o u r tout
x*~X ~ , x~(X) = lim x (x n) ; or
mesurable
~(~o
I-I). On a 2 a = ~
x~(x~) est mesurable ~ n
= ~(~o
) ' car
' doric x~(x) est
) et ce raisonnement vaut pour route fonction continue.
Le lemme de Prohorov pourra s'appliquer ~ o
des cylindres{ x : [IXnll ~ 11 qui
:
Soit ~ o ¢ dams
X
est l i m ~
~o
au lieu de ~
(cf. (5)), en particulier
est f~m~lle d'umicit@ pour route mesure tendue d@finie sur J~.
3) Les x (1) constituent ~videmment une base faible de X*, et une suite s@parable pour les points de X puisque x(i)(x) = 0, tout i
x = 0.
4) X est s6parable mais non n~cessairement m@trisable. Exemple L'espace
:
des distributions de
L. Schwartz (sur
R
ou
Rn ). (X, ~ )
peut n'@tre pas
localement convexe, mais si on r@duit ~ a la topologie faible ~ (X,X ~) sans changer X ~ (par d@finition) doric sans toucher ~ e t
~ la continuit@ des
x (i)
, (11)
est
-
conserv6 et
(X,~)
318
-
devient localement convexe ( on d i m i n u e ~
2)
Dams ce qui suit, nous rempla$ons les projections
les
Pn : n = PmP
py
).
des parties I e t
IV par
'
la compatibilit@ d'un syst@me
{PnJ
signifie que
m > n @ Pm n P m
=Pn
' et le lemme 2
et ses corollaires valent toujours. Notre probl@me comcerme essentiellement les lois vE v m = e(Fm) : d@finir
F
telle
que ~ = e(F) . Lemme 12
:
Si V , ~-temdue,
r@guli@re, d@finie sur ~ , a des projections
Fm
Em-0 )
(port@e par
et ~ -temdue (+)) Fm
dams les
(12)
~
~ m = e(Fn)' ~ chaque
@tant born@e et sym@trique, il existe
d@finie sur ~ et port@e par
F
unique ( sym@trique
X-O, et de projections
Pm
F
@gales
~
F, telles
Em-0 , telle que : = e(~).
(12) signifie ici qu4il existe une suite de mesures ~ - t e m d u e s born@es que lim e(~ m) = V . D@monstration :
I1 est essentiel de moter que les EFnU me somt pas @gales, peuvent croitre infiniment avec
n , parceque chacune est port@e par
en d@signant par
E'm,n la partie
AS "Relevons" succesivement des
Fn
fl = [FIS
p~
(E~-0)
. Les
de
e(Fln)
le lemme 2, @galent
F1
En-0
(P~n I0} ) dans
,donc de
E n,
que
[Fn]
- [Fn] = Fn,('n,n)
qui se projette en
0 sur
E n.
E2,...En,... , c.O.d, consid@rons les restrictioms
E n ; ces restrictions
F1
sont de meme variations totales
sont lois sym@tris6es (++~ , facteurs de ~ n = Pn v doric, vu
Pm e(~l)
= e(pn ~i ) avec
Pn ~1
= ~In ' ~1 a ~ X ' ~ ) "
(+) en ce sens qu'elle est limite d'une suite croissante de mesures born@es ~-tendues. (++) car
Fn = ~
(Fm + Fn).
-
319
-
fl En effet
A ( 5Bn, A o K ~ , n = ¢ @ e ( F l n ) ~ A ) < E
B] Recommen~ons ce rel~vement masse ~1
F2 - F1 2
E~ 1
~
Fln A < ( e
(pattie de
de variation totale
E2
•
port@e par l'axe
% e2) , pour la
f2 - fl ; on en d~duit de poche en poche, avec
= G1 : N
(13)
~ = e(G1) .... e ( G N ) ~ N
On a
Pn YN = Fn
pour
= e(~N) ~ N
N~ n , dams
est U.T., sans translations, les
' avec ~
= Z
G n e (X, JB)
En-O. La convolution (13) , enlev~ la facteur ~ N '
e(GN)
~tant des lois sym~tris~es, Elle converge vers
puisque toute loi limite ~' pour cette f~m~lle U~T. a pour projections ~ n' = e(Fn) :
L'unicit~ de
F
Th~or@me,III-1
: E n , de ~ loi @ -tendue sur (X,J~), sont du type de Poisson
au sens de (3)), on a V = e(F)
existe une f~mille filtrante~ de sup i
n
r~sulte du lemme 3 et de la remarque ~ du n°III-1 .
Bi les projection ~ n ' sur (V n = e(Fm)
V
F i = F , et que ( pour des
Fi ai
(et ~ )
, en ce sens (lemme 10) qu'il
born~es, ~ -tendues, d~finies s m r ~ ,
telle que
convenables) la famille U.T. { e(Fi)ai I a pour
seules lois limites des translat~es de v . Si
X
est m~trisable ( ou si le sont les
KE
sous-tendant v
) on peut choisir les
de fa$on que
e ( ~ ) a i converge vers v, suivant le filtre croissant d~fini par les
D~monstration
:
A] Soit
Ui
nage de
0 ).
On a i
~ U ' D U , pour chacun de ces ntoo
~tant fix~, "relevons" la restriction
p~l I
F~ i
~(+~
un voisinage (par example ouvert) de 0 (ig I, I indexe une Oase de voisi-
U n = pnl{p n U}
vante :
ai
~
de
F1
U
(point important pour la suite).
~
El-P1 U i
s'obtient en ajoutant, darts E2, ~ la restriction de
( E l - P 1 U i ) ) , la pattie de
F2
port~e p a r % e 2 (soit
, de la fa~on suiF2
x~= 0 ), et ext6rieure
(+7 Ii en est probablement ainsi darts le cas g~n~ral (compte tenu de l'hpoth~se d'une base) •
- 320 -
P2 Ui ; de m~me dams vant
E3
F~ i , la partie de
compatibles) pour chaque
on ajoute ~ la pattie de F3 i
ext~rieure ~ . et on a
F3
qui on projette dams
P3 Ui ' et port@e par ~ e 3. Les
i
"
sui-
F~ i
sont
sup F~ i = Fn .
(i fix@) d@finissent F'$6(X~) car la sym@trisa~ion de ~ ni =
BS Les F'~•
~
(~±)
donne
°-
d'apr@s le lemme 2 des$ n s=e (F'~ + F'~n ) projections d'une loi r i (facteur de ~) " . , et suivant le lemme 12, les F'~ + F'~- sont projections de F 'i + F'i-6 (X,~5), born@e hors de tout
( v u l e lemme 7), en fait ext@rieure
P- (Pn Ui> =
U i,
n
nulle dams
Ui
. Ainsi
Fn i = Pn F'i Puisque Les
Pn e(F'i)
Vn
born@e, nulle dams
pour tout
n ,
e(F 'i)
U iest facteur de
F 'i forment une famille filtrante croissante (en fait
striction de
F 'j
sup i Les
divise
' F'i
~ son support
F 'i = F E ( X , ~ B )
e(F'l)a i
sont U.T.
,
A i ext~rieur ~ et
(pour des translations
K ~i
(pour des ~ ~
, Aic Aj ), donc
ai
convenables ), et les lois l~m4te~
9 et 10). I1 est @vident (lemme 10) qu'il
est de meme pour toute autre famille filtrante
ces m~mes
e(Fi)al tels que ~ i
F i ~ F.
sont
( ai
convenables ) sous-tendues par
< 1 ), ces lois appartiennent d u n e
famille compacte m@trisable, et une adapt ation (au cas de la CV. suivant la d@monstration du th@or@me II de [11] I
~
F ) de
compl@te la preuve du th@or@me (pour le cas
non d@nombrable). Ajoutons que les lois limites pour les
translations les rendant U.T.
est la re-
corollaire 1 du lemme 2 puisqu'il en est ainsl
en projection (ou encore d'apr@s les lemmes
Kg i sont m@trisables, les
F 'i
pn F = Fn .
@galent ~ ~ ume translation pr@s, v u l e
Si les
Ui , ~
Uj~ U i ~
~ (lemme 2) .
e(F'i/n), n fix@, apr@s
(on v@rifie que ce peuvent ~tre les
ai/n), ont pour
n-i@me puisaance une translat@e de ~ , on a bien ~ a g , Par contre nous me savons pas montrer que ces v l / n
sont U.T., encore moins que ~ t existe et est continu.
Corollaire : Si les projections
PnP = ~n
de p tendue sont ind@finiment divisibles, p
l'est (avec
-
1In
321
-
tendue)
Th~or@me I I I -
2 :
Dams tout espace vectoriel topologique (s@par~) admettant une base, s i p une loi ~ -tendue ( ~ topologie compatible avec la dualit~ d@finissant
~ (X, ~ ) est X~
et~=J~
a)
ind@finiment divisible, on a : a) 0
I1 existe une mesure
F tendue unique ~ (X, ~ ) born@e hors de tout voisinage de
, et une d6composition p = 2 ~
,(~ et ~ tendues ) unique ~ une translation pr@s satis-
faisant
(14)
b)
~
=~9
, les
x~(~)
sont n o r m a l e s , ~ = e(F) ( cf. lemme 10 ou th. III-1 ).
cette factorisation est ( ~ une translation pr@s) l'unique d@composition od
serait seulement assujetie ~ appartenlr ~ S e t e(G) c)
~ n'avoir aucun facteur propre du type
(suivant le th. II). Par projection sur
Y=X/X o
(Xo sous-espace ferm@ de
d@composition correspondante de ~ y a S (Y) l'unicit~ de
Fy
; lorsque
(pIus large)
et
Fy
1
X ), p y
= ~y ~y
est la
r@pondant ~ a ) sauf en ce qui concerne
a une base, c'est la seule d@composition au sens b)
est unique ( = S y F).
D~monstration : a)
r6sulte du corollaire ~ du lemme 2 et du th@or@me III-1 , en notant que chaque
x~(x)
~tant limite (partout sur
X) de
x *(Pn x), x ~ ( ~ )
est une fonctionnelle lin@aire continue de
est normale, car x (Pn x)
Xn = Pn x : la restriction de
xs ~
En,
de loi ~videmment normale. b)
S o i t ~ = ~' ~' , ~' = e(F') , ~ ' ~
~(X)
et n'ayant pas de facteur du type de
Poisson. Les projections ~n
=~n
e(F~)
Pn(F-F')
@galent
est une d@composition de ~ n
n'§tait pas nulle, pour tout par les
Fn-F'n = F n'' dams
n , on aurait
En-0
et
y
sont ~ 0
dams la classe ~ ( E n) F''
F~' sur les ~ n (cf. lemme 3), et v u l e
; si
~tant bien d@finie s u r ~ lemme 2 :
car F~' (X-O)
-
~ n = I~' e(F~')
~
~n' = ~ n
e(Fn'' ) ~
322
-
~' = ~ e(F'') (sur ~ )
d'od la contradicti-
on=
c) ~y
= ~y ~y est, darts ~(Y)
et l'unicit@
une factorisation de ~ y
( ~ une translation pr@s ) de 2y, ~ y
les m@mes Justifications
qu'en a). Lorsque
cises d~ c ) r@sultent aussi de
Y
: elle satisfait @videmment ~ (1@)
(avec V y = e(Fy) , Fy = W y
F) a
a une base les conclusions plus pr@-
a).
--IV--
Cas od
X
t localement convexe et m@trisable t e s t espace de Badr~k~an.
Lemme I~ : Supposons qu'il existe dams S
X~
dual de
(X,~)
( X
localement convexe) une topologie
, telle que
a) l a ~ S
continuit@ de la fonction ~(~)
est condition n~cessaire et suffisante pour
que cette fonction suppos@e de type positif ( avec ~ (0) = I ) soit fonctionnelle caract&ristique
(f.c.) d'une loi ~ -tendue (r@guli@re sur (X, ~5 ),
b)l'@quicontinuit@
d'une f~m~lle ~
lois p~ correspondantes (r@guli@~e, (15)
p
est condition suffisante pour que les
soient U.T. ; alors route loip de ~(x) (au sense1)) ~ -tendue
d@finie surds ) satisfait = lim e ( n ~ I/n) , et
D@mons~ration p 1/n
(pour ~S)
pl/n
~ n--= oo
O.
:
est unique e t a
pour f.c.
I/n(
~0,
si ~ ( x ~)
est f.c. de p (n6cessairement
partout ~ 0 ). Soit Posons
~
tel que
u = 1-~, ! n Log W
x~
US==)> 11- ~ ( x ~) I~£ •
on a (pour tout r ul( ~ )
= u
e
H + H
u2
, IeI%1 ; l e s
e(n
pl/n
)
ont pour deuxi@me f. c.
-
n ( ~l/n-1)~n
Log
que les
x~(~ )
Remar~ue
:
~I/n~u
@galent
, uniform6ment x ~(p),
US
~cc
p t-*o
0) :
:
(Ki, i ~ I) de parties compactes
tlennes. Cela signifie que tout compact
Les
R n.
assurant a) et b) est la suivante
admet un syst@me fondamental
(pour chaque
u s ,ces lois limites ~ sont telles
p t, t > O, est bien d@fini et continu.
L a condition de Badrikian,
sont hilbertiens,
dams
) sont U. T. , on peut affirmer que
(car les lois limites sont , en projection,
X
-
car (15) vaut dams
l~misque, aussi bien, les pt (0 ~ t ~1
le semi-groupe
323
K
, convexes hilber-
est contenu dans un
c.O.d, que le sous-espace norm& de
X
Ki
, et que les
d@finit par U xll =
Ki
inf l~l
K i) hilbertien,
(base de voisinages
ouverts de 0 p o u r ~ S )
sont des ouverts pour la topologie
de CV. umiforme sur les compacts convexes de
X (~S
~cc
)"
Th~or@me IV : Si
X
(localement convexe)
est m@trisable
lemme 13, route loi p 8 - t e n d u e
de 7(X, 2 )
et satisfait aux conditions a) et b) du (+) admet dams 2 u n e
d@composition ~ = 9 e(F)
unique ~ ume translation pr@s telle que : a)
les = lim
b)
x~(~ )
sont normales, ~ n'a aucun facteur (propre ou non) du type
e (Gn) avec
G n U n~- ~ c o
cette d~composition
on correspondamte
dams
0
pour tout
e(G) ,
U ,
se projette sur tout espace
Y = X/Xo,
suivant la d~compositi-
Y .
La preuve de a) r~sulte du th~or@me II, de (15) et du n°II-4
; celle de 0) de ce que
satisfait aux m~mes hypothe"s es. On notera que suivant II-%, et le corollaire 2 du lemme2, les trois caract~risatiens (+)
au sens de (1),
1/n
est alors temdue (d'apr@s le lemme 13 ).
Y
- 324-
suivamtes de ~ normale sont (sous les hypoth@s@s les x ~ ( 9 ) ~ 7
(doric ~ e S
L'hypoth@se
route loi tendue F
(alors
~
a
~
),
et n'a pas de facteur e(G),
oo ~ ( 1 0 ) Remarque:
sont normaSes
a) et b) du lemme 13) @quivalentes:
~e
~
).
peut @tre remplac@e par
dont les p y e
~ (Y)
~ye
est ind@finiment
~(Y)
divisible
(tout Y e
~ ):
( vu la d@monstra-
tion du lemme 13) sous les conditions de cette pattie IV.
-VX est un espace polomais de dual faiblement s@parable.
V-l: Rappelons
que darts un espace polonais
est tendue.
~ Yn' n = 1,2,...)
Hest
~-~
L'application
un sous-groupe
D@monstration: ~-
toute loi sur
(cf. ~I@U lorsque X est hilbertien s@parable)
le
~ de X darts Z = T °o d@fimi par x
Lemme i4:
et complet)
@rant une suite de X ~ dense dams X ~ p o u r la topologie
faible ( ~ (X~,X)) de X ~, consid@rons plongement
(s@parable, m@trisable
est biunivoque
~
z =
I z(n) ,
~ est continue,
de Z, bor~lien
est continue,
n = 1,2,... ) et biunivoque
dans Z
, z(n)
(de tribu bor@lienne
.
~').(++)
(produit)
de T °~.
du groupe X dams le groupe Z et z(n)(x) = 1,
tout n? ~=~ x = 0 car les Yn(X) doivent approcher arbitrairement donc chaque ~ x~(x) ce qui, si yn(x) -- 0 ( 2 ~ ) de Z e s t
ei yn(X)
entre X et son image ~ X = H.
les z (n) l'@tant, pour la topologie
car est un homorphisme
chaque x~(x) = O. H, sous-groupe
=
et
chaque x~(x),
x~(x) ~ 0 est impossible,
bor@lien (H ~ ~ ' )
continue injective de X polonais darts Z m§trisable
(x fix~)
donc
dams Z car application
(cf.Boubaki,
[3~ paragraphe 6, Th.3).
fermeture de H dams Z est un groupe compact qui peut remplacer Z dams ce qui suit. V -2: Soit
~ ~ (X, dS).
~
, son image dans H, est la restriction ~ H d'une loi p
de ( Z , ~ )
- 325 -
caract@ris6e par Lemme 1~:
~H = I.
Toute factorisation,
~ : ~ ~ , dams Z de ~ image d e p
translations inverses) g une factorisation dams H. S i p
, se ram@me (par
a U (X), ~ e U(Z)
et r@cipro-
quement si p'H = 1. Demonstration: I°)
vu
~
=
J~(z
-1 H) ~(dz) = 1,
~z(~_
eg_
~(Z -1 H) = 1, 1 = ~-H -- f f
tel que
(suppox~ de ~) v
soit z z-I
~(H) = 1, doriC, vu
(z '-'1 H)z
~(dz')~
~ = ~z -1 z~ ,
~ z-I(H),
H donc
~ z-1 est aussi "pottle" par H.
2°)
Seule la r6ciproque demande v@rification: si ~ = ~ n Zn [8] que zn me d@pend pas de n ) (w ~ n )n z 'n n
=
Lemme 16:
Zn ~
Z
on a, vu 1°),
,n n
Zn~H
~
~W
~n = r~n ~ ~(H)
sont "port@es" par
tears (propres) Lemme 17.
z' n d'ou 4-->
~ ~ ~(X)
Si ~ c ~ (X) n'a pas de facteur propre e(F), il en est de mSme de ~.
Si e(~) @tait un tel facteur de ~, il existerait z tel que z et ~ z
(en fait, on salt, d'apr@s
H~
"Poisson", v u l e
Toute l o i ~
z g H ~
~ e(~) z~
~ H = 1, et ~-~ donc ~
(Z-H) = O, doric auraient des fac-
lemme pr@c@dent.
normale au sens du th~or~me I I a ses x~(~ ) normales.
Demonstration: C'est celle de [14] , pour montrer que
~
~ a ses x*(.) normales (ce qui suffit) s
est, dams Z, normale au m~me sens, donc d'apr@s ~8~ , sa fonction caract~ristique, finie sur le groupe @+) Les
e
des caract@res e(z) = (z(1)) nl .... (z(k)) nK
d@-
de Z, est
points qui suivent (lemme 1@ ~ 15) sont implicites dams E'I@] mais non pr6cis@s.
on peut montrer que lorsque complet l'image de X est ferm@e ).
un
X
est espace de Banach faiblement s@quentiellement F~
dams
Z
( l'image de IIx II ~
1
est
- 326-
od
Q(e)
est ~
O, sym~trlque et satisfait K:
Q(ee,) + Q(ee ,-I) 9
(x*)
E {ei x ~ ( ~ ) ~
~tamt la f.c.
~e
161 reals
=
~ (x ~)
lq(x
=
8) I
?-K
2 [ Q(e) + Q(o,)]
=
de ~
, toujours
.
~ O,
on a pour
=
Q(e)
~ ~ e
soit
~
Log
?
(x e)
-
est s g q u e n t i e l l e m e n t f a i b l e m e n t c o n t i n u e dans X*
(x~Cx) --~
x~x)
~-~
ei x ~ ( x ) _ _ ~
ei x~(x)___~, ~? Cx~)
>
?(z'D),
on en d~duit par continuit~ que:
(x')
D'od
~(t x ~)
Remarque:
=
=
-Logl ~(x*)l ~ o e
...
=
2 [ W ( x ~) + ~ ( x , * ) S
.
t 2 ~ ( x ~)
W (x~ + xl~) + ~ (x~ - x'))
et x~(9) eat normale, sa loi sym~tris@e l'$tant.
Ii parait difficile d'exploiter, par cette m6thode, les connaissamces (de
[8] ) sur ~(Z). Par exemple que
(pour ~ a S(X))
on a ~
=
lira e ( n ~
I/n), au sens
de la CV. des lois aAns Z. V - ~"
Th@or@me V-
Dams route espaae polonais, dont le dual faible est s6parable, la d@composition ~ = 2 ~ du th@or&me II de ~ g ~(X) est unique g une translation pr@s et telle que: a)
les x~(9)
sont normales,
b)
la mesure F telle que h~ = e(F) est 1~n~que,
c)
pour tout sous-espace ferm@ factorisation,
Xo, la projection sur Y = X/X o de cette
est l'unique factorisation,
(~ une translation Dr@s), r6-
pondant dams Y au th@or@m II. D6monstration: a)
r&sulte du lemme 17, l'unicit~ ~ une translation pr@s du lemme 2, corollaire 4,
-
327
-
b)
du lemme 3
c)
du mSme coroXlaire @ et du fair que Y est espace de Fr~chet, s@parable car g base
(ici ~
=4
),
d@nombrable de voisinage comme projection de X qui est tel, et que merit s@parable pour la m~me raison
Abr6viations et notations.
=
X-A
tribu bor@lienne
fG ~ = ~(x) fG
f continue born6e sur X f~ ~ et est ~ - m e s u r a b l e
~'
tribu cylindrique ¢ Y
=
~ (x)
p
ind~finiment divisible
pattie r@elle de variations totales de p , F. Y CV.
est faible-
(comme sous-espace de X ~ faible g base
nombrable de voisinages ).
A c
Y~
espace quotient de dimension finie
converge, convergeant,
...
f.c.
fonctionelle caract@ristique,
F
ensemble f e ~ ,
K
ensemble compact
U.T.
uniform~ment tendu (@quitendu).
Voa.
variable al@atoire.
d@-
-
528
-
Bibliographie (1)
Ahmad
Annales de l'Institute H. Poincar@
(2)
BadrikianA.
Th~se
(3)
Bourbaki
Topologie chapitre IX.
(4)
Csisz&r I.
On infinite products of random elements and infinite convolutions
(Paris)
(1965).
~ para~tre.
of probability distributions on locally compact groups. A para~tre darts Zeitschrift - Wahrscheinlichkeitsrechnung.
(5) (6)
G r e n ~ d e r U.
Probabilities on algebraic structures - Stokholm 1963
Mahlon M. Day
Normed linear spaces - Springer(Ergebnisse
(7)
Parthasarathy, Ranga Rao and Varadhan
21) 1962
Probability distributions on locally compact abelien groups. Illinois J. of Math. 7
(8)
Mourier E.
(1963) PP.337-369
El@merits al@atoires darts un espace de Banach. Annales Inst. H. Poincar@ 1953.
(9)
Parthasarathy and V.V. Sazonov On the representation of infinitely divisible ddstributions on locally compact abelien groups, Theory of Probability and its applications.
(10) Tortrat A.
IX (1964) pp. 108-111.
Lois de probabilit@ sur un espace topologique compl@tement r@gulier et produits infinis ~ termes ind@pendants dans un groupe topologique. Ann. Inst. H. Poincar@ 1 (1965) pp. 217-237.
(11) Tortrat A.
Lois tendues et convolutions d@nombrables dams un groupe topogique
(12) Tortrat A.
X . Ann. Inst. H. Poincar@ (1966) pp. 279-298.
Lois ind@finiment divisibles dams un groupe topologique ab@lien m@trisable
X
, cas des espaces vectoriels.
C.R.A.S. 261 (1965) p. 4973. (13) Urbanik
Studia Mathematica
1960
pp. 77-88.
(14) Varadhan S.R.S.
Limit theorems for sums of independant random variables with value in a Hilbert space. S~n~hya 24 (1962)
pp. 213-238.
-
329 -
On a paper of J.G. Sinai on ~ynamical s2stems. Heiner
Zieschang
It was given an information lecture on Sinai's paper "Classical dynamical systems with countable Lebesgue spectrum. II", Iswestya 30, 15 - 69
(1966). The main concepts were
explained in detail, the hints for proofs were poor. Furthermore things were illustrated at the example of a continuous automorphism of a compact connected Lie group G. Already here the main ideas can be shown and no technical troubles come in. From the results of J.G.Sinai follows: If the induced linear mapping A = (dT) e on the tangent space E at the unit element has no eigenvalues of module 1, T is a K-automorphism and has therefore all nice mixing properties with respect to Haar measure. For simplicity let uS assume that eigenvectors v 1 ,... ,vn with eigenvalues 4 1 ,..., ~ n ( I ~ i [ < 1, i ~ k, [ ~ i ~ ~ 1 ,
i > k)
span the tangent space. We introduce a metric on E
b y defining v I , ..., v n orthogonal and of equal length and transport this all over G b y left translations.
This gives a left invariant Riemamnian metric on G. The length of
the v's is chosen so that the total volume of G equals 1. Let V be the linear subspace, spanned b y vfl, . .. ,Vk, W spanned b y Vk+ 1 ,... ,vn. By the exponential mapping we project the parallels to V resp. W down to G and obtain two measurable foliations,
expanding
resp. contracting transversal fields for T amd absolutely continuous one with respect to the other. Thus all conditions needed in Sinai's theorems are fulfilled. It is clear that T represents an "U"-cascade in the sense of D.W. Amosov (Soviet Math. Doklady @ N ° @
(1963),
1153
- 1156).
