Geometrical and Statistical Aspects of Probability in Banach Spaces

I II),

that: 3: K > 0:

'IkE N,

a. s,

Let's define for every integer n and every £ E B' : A(n, f) = 2 -2n I: E f 2 (X ) k k E I(n) Suppose that assumptions a), b),

cl

of Theorem 1 are fulfilled and that the follo-

wing one holds also:

V

d)

E

> 0,

V fEB' ,

exp ( _ E

I:

I

A(n, f))

<

+

til ,

n :2: 1

Then:

In ....

p( w : S (w ) n

0 weakly)

= 1.

PROOF: By Theorem 1, we know that:

p(

sup

II

n

S

n

In

I

<

+

al )

=1

A standard argument then gives [13 J E

sup

I

Sn

In

I

<

+

(7)

al

n

Furthermore the one dimensional Prohorov SLLN [20 'If E B',

f( Sn

In)

....

0

J implies

a. s.

This property and (7) show that (Sn

In ,

0(X

l

, ..• , X ) ) is a weak sequential n

amart of class (B) (for the definition and the main properties of weak sequential amarts, see for instance [4J V.3 ),

37

The space B being reflexive, the conclusion of Theorem 2 follows immediately from a well known convergence theorem of weak sequential amarts due to BruneI and Sucheston [2 J.

§ 4. PROHOROV'S STRONG LAW OF LARGE NUMBERS. The sufficient condition for the SLLN in the Prohorov setting can now be guessed easily from Theorems 1 and 2 ; the statement is as follows : THEOREM 3 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real, separable, p-uniformly smooth ( 1 < P " 2 ) Banach space (B, II II ), such that: 2 K > 0

E i'-I

11 k

a. s.

Suppose that the following hold:

In

....

a I)

Sn

b ')

2-2np Log n

c')

11

in probability.

0

II X

I:

k E I(n) E

>0,

I:

exp (-

k

I

E

11

2p

i\.(n) )

.... o in <+

til

probability .

•

n :2: 1

Then: S

n

In

->

0 a. s.

REMARK: It is easy to see that b ' ) holds for instance if : n - 2p L n I: II X 11 2 p .... 0 in probability. k 2 1 ~"n PROOF: By a classical argument [17

J

it is sufficient to consider the symmetrical

case. By symmetrization ( see [10 J proof of Lemma 1 ) b') implies: b ll )

lim 2-2np Log n n .... + ~

E II X

I:

k E I(n)

k

fP

:: O.

The ideas of the proof of Theorem 3 are the same as those used for proving Theorem 1 ; so we will keep the same notations as in the proof of Theorem 1 and we will only detail what needs to be detailed. First one notices [1 EIITnllp ....

J

that from a') it follows that: E II Sn

In

liP .... 0 , and also:

o.

So the the conclusion of Theorem 3 will be true if we show: Vx >

I:

0,

n :2: 1

p( II T

n

liP - E II T

n

liP> ?,x)

< + dl •

Without loss of generality it suffices to check (8) only for x E

(8 )

J 0,

243- P [ • Fix

such an x. As previously, we denote by u we will bound u

n

n

the general term of the series involved in (8) ;

by considering again two classes of indices n, slightly different

38

from those taken in the proof of Theorem 1. Let's suppose - and this of course isn't a loss of generality - that: n 'In:2:2, 'ljEI(n), Ilx.II,,2 /Logn a.s •• J For simplicity we put 8 :: exp ( -8(2p+2C ) Ix) p

We consider fir st the following situation: I\. Log n" 8.

First case:

It is easy to see that if one puts y :: 243 x

l/p

, relation (6) holds for n large enough.

Now we will again apply Lemma 2 ; for this we introduce the following sequence ( Yk ) :

'l k :: 1, • • •. 2

n

One has: 'l k :: 1, .. "

2

n

I "

Yk

1 a. s.

and:

VI

+... + V 2

where:

d

"

n

n

....

(2(Log n)

2

I\. + 2Log n d

n

),

0. by condition bll) .

Therefore:

V n :
Vl+•.• +V n

,,48Logn.

2

Applying now Lemma 2 with v:: 4 8 Log nand A :: 4 v (2p+2C ) p 3/2 u s: ( 2/ch2Log n ) exp 4 81 12 Log n ,,4 nn

It remains to consider the complementary situation:

A Log n > 8 .

Second case:

If one choos es now:

Y

:: ( 8

k

I

I\. (2p+2C) P

) Sk

one has: 'l k :: 1. ... , 2

n

,

I Yk I

,,1

a. s.

and: 'In:2:n'(x)

VI +... +V

n

"48

2

I

1\..

2

Applying finally Lemma 2 for v :: 4 8 2 u

n

,,(2

I

I

ch(x387/8/21\.)) exp (4x482/Al

I\. and

A:: v x

2

one gets 3 7 ,,4 exp (_ x 8 18 141\.)

Collecting the partial results we obtain: 'l x E JO, 243- P C, II N(x) EU'J, II 'Y (x) > 3/2 'l n:
°:

and (8) immediately follows from hypothesis c ' ) •

lx,

one gets:

39 REMARK: Hypothesis b I) in Theorem 3 seems at first glance somewhat surpnsmg and artificial. To shed light on its meaning we will now give some corollaries of Theorem 3 which will show that b l ) is a very weak hypothesis.

COROLLAR Y 1 : Let (X ) be a sequence of independent, centered, r. v. with k values in a real, separable, p-uniformly smooth ( 1 < P " 2 ) Banach space (B, (B,

I II),

I II)

such that:

I

:3: K > 0 : Vk E N,

Xk

II " K

a. s.

(k/L k) 2

Suppose that the following hold: 1) n- P I: 1/ X liP .... 0 in probability. k 1 "k" n 2)

'l

>0

E

I:

exp (-

E

I

<+~ •

1I.(n))

n ;;, 1

Then:

a. s.

The proof of this result is very easy. First: 2-2np

II

I:

,, (c/Log n) 2-2np

I/2p

X

k E I(n)

I:

I

X

k E I(n)

k

k

liP

and so hypothesis b') of Theorem 3 is fulfilled by applying 1). By symmetrization one has n -p

I:

1

~"n

E

I

.... o

liP

X k

the space B being of type p, it follows that the sequence ( Sn

In )

converges to 0

in LP(B). All the assumptions of Theorem 3 being fulfilled, this ends the proof of Corollary 1. In the case p :: 2, Corollary 1 reduces to Prohorov's SLLN proved in [11 J

•

Let's

stay for a moment in this situation p :: 2 for making precise the difference between the hypotheses of Theorem 3 and those of the result in [11

J•

One observes first

that the following easy corollary of Theorem 3 holds: COROLLAR Y 2 : Let (X ) be a sequence of independent, centered, r. v. k values in a real, separable, 2-uniformly smooth Banach space (B, 1/

II),

that:

II

:3: K > 0: 'l kEN.

X

k

"K (kl L k)

1/

2

a. s.

Suppose that the following hold: a') b ,)

Sin n

( 1

c ,) 'l

I

n

2

.... 0 in probability. L n ) 2

e: > 0 ,

I: 1 "k", n I:

I

X k 1/

exp ( -

E

2

I

.... o ~robability 1I.(n))

< +~

n ;;, 1

Then:

Sin n

.... 0 a. s.

•

with such

40 The gap between hypothesis b ' ) in the above statement and assumption 1) in Corollary I is clear. It is easy to give examples of sequences of r. v. which belong to the domain of application of Corollary 2, but not to the one of Corollary I ; the sequence considered in the last section of [18 In [II

J

J provides

such an example.

it has been noticed that in the special case of Hilbert spaces - which

are of course 2-uniformly smooth - condition 1) in Corollary I is necessary for the 5LLN in the Prohorov setting. More generally, is it possible to simplify the hypotheses of Theorem 3 by making additional cotype restrictions on B ? Godbole has characterized the spaces of cotype q in terms of 5LLN for the sequence ( II X

Ilq / kq-l) ( [6J Theorem 2. I ) ; unfcrtunately his result will of no k help in our situation. The ( partial) result we are able to prove is as follows COROLLARY 3 : Let 3/2 ,;;; p,;;; 2 and 2 ,;;; q,;;; 2p - I ; consider a sequence (X ) of k independent. centered, r. v. with values in a real, separable, p-uniformly smooth Banach space (B,

I Ill.

of cotype q.

Suppose that the following hold: i ) 3: K > 0 ii) V

€

V kEN

> 0

II X k " exp ( -

L

€ /

,;;; K (k/ L k) 2

<+

A(n))

a. s.

l!l

n ;;, I

Then:

5

o a.

/ n

n

s.

5

n

o in

/ n

probability.

PROOF: Of course the only thing to do is to check that the weak law of large numbers implies the strong one. As 5

n

/ n _

0 in probability, it is sufficient to consi-

der the symmetrical case. By cotype q and assumption lim n _ +

n -q

Hence: -2p L n 2n

II

E

L

I s:k:s:n

!II

L

ls:k,;;;n

E II X

k

n,

one has: q :: X Il k

11

O.

2p ,;;; K 2p - q n- q (L n)q+I-2 p 2

L

Ell X

l,;;;k,;;;n

k

Il q

,

and Corollary 3 immediately follows, by application of Theorem 3. REMARK: Let's suppose that conditions i) and ii) of Corollary 3 are fulfilled for a sequence of symmetrically distributed r. v. X • If the weak law of large numbers k q holds, then, by Godbole's result nL II X Ilq 0 a. s •• So the following k implication holds : l,;;;k,;;;n q 5 / n - 0 in probability => n- 0 a. s. n this shows the very special geometric nature of these spaces which are both p-uniformly smooth ( 3/2 s: P ,;;; 2 ) and of cotype q

(2,;;; q s: 2p - I ) •

41

§ 5. APPENDIX

SOME LAWS OF LARGE NUMBERS OF KOLMOGOROV-BRUNK

TYPE.

In [9J and [12J the SLLN of Kolmogorov and Brunk are studied in 2-uniformly smooth spaces under hypotheses requiring both assumptions on the strong and on the weak moments of the r. v •• Results stronger than the classical ones known for type 2 spaces are obtained. By applying - in a more elementary manner as above - the martingale technique developed in section 2, Kolmogorov-Brunk type SLLN can also be obtained in p-uniformly smooth ( 1 < P " 2 ) spaces; in the case p = 2 these results are stronger than the ones given previously in [9J and [12J • The proofs being elementary, we only state the results.

THEOREM 4 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real, separable. p-uniformly smooth ( 1 < P " 2 ) Banach space (B,

I II).

Suppose that: i ) There exists a sequence of positive numbers (d ) , converging to 0, such that j "if j EN. Xj " j dj a. s.

II

ii)

I

Sn / n .... 0 in probability.

1£ one defines for every integer n : r(p,n) = 2-2np

Ell X. J

I:

j E I(n)

11

2p

then the following implication holds

( 3: k integer, k;" 1:

(l\.(n)

I:

+ r(p, n))k < + ~ )

n ;" 1

=>

S

n

/ n

.... 0 a. s.

As an obvious corollary of Theorem 4 one has the following result which can be compared with the well known law of large numbers in type p spaces of HoffmannJ~rgensen and Pisier [15J

COROLLAR Y 4 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real, separable. p-uniformly smooth ( 1 < P " 2 ) Banach space (B,

I II).

Suppos e that: a)

Sn / n .... 0 in probability. I:

b)

k- 2p E

k;" 1 c)

3: j ;" 1 :

I:

I

Xk

2P 11

( !(n) ) j

< + <+

~

~

n ;" 1 Then:

S

n

/ n

.... 0

a. s.

42

Bounded laws of large numbers and laws of large numbers in the weak topology can of course also be considered in the Kolmogorov-Brunk setting. We only give an example of such a result : THEOREM 5 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real separable. p-uniformly smooth ( 1 < P ~ 2 ) Banach space (B,

I II ).

Suppose that: a) The seguence ( Sn / n ) is stochastically bounded. 2p b) I: kE X fp < + ~

I

k :2: 1

c)

3: j

:2:

1:

k

I:

(A (n))j

< + (fJ

II

I

•

n :2: 1

Then:

p(

sup n

Sn / n

<

+al)

=

1

Let's conclude by mentioning an application of the above results. It is well known that all the classical situations in which one can conclude that a Banach space valued. symmetrically distributed r. v.

X - and it is always possible to

reduce to that case - satisfies the LIL. are the union of a Prohorov type BLLN and of a SLLN ( [16 J [8 J

•

.

[7 J

) or

of a law of large numbers in the weak topology

Therefore it is not surprising that Ledoux's recent necessary and sufficient

condition for the LIL in uniformly convex spaces [19J can also be obtained as a corollary of Theorem 1 and Theorem 5. The computations for proving this observation are left to the reader. REFERENCES.

[lJ

DE ACOSTA, A. : Inequalities for B_valued random vectors with applications to the strong law of large numbers. Ann. Prob. 9 (1981), 157-161

[2J

BRUNEL, A. et SUCHESTON. L. : Sur les amarts faibles a valeurs vectorielles. C. R. Acad. Sc. Paris 282, Ser. A (1976). 1011-1014

[3J

DUBINS. L. E. and FREEDMAN. D. A. : A sharper form of the Borel-Cantelli lemma and the strong law. Ann. Math. Stat. 36 (1965). 800 -807

[4J

EGGHE, L. : Stopping time techniques for analysts and probabilists. Cambridge University Press - Cambridge 1984

[5 J

ENFLO. P. : Banach spaces which can be given an equivalent uniformly convex norm. Israel J. of Math. 13 (1972). 281 - 288

[6 J

GODBOLE, A. : Strong laws of large numbers and laws of the iterated loga_ rithm in Banach spaces. PHD. Michigan State University 1984

[7 J

HEINKEL, B. : Relation entre theoreme central-limite et loi du logarithme itere da.'1s les espaces de Banach. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49 (1979). 211-220

43

[8 J

HEINKEL. B. : Sur la loi du logarithme Here dans les espaces reflexiis. Seminaire de Probabilites 16 - 1980/81 - Lecture Notes in Math 920. 602-608

[9J

HEINKEL, B. : On the law of large numbers in 2-uniiormly smooth Banach spaces. Ann. Prob. 12 (1984), 851-857

[lOJ HEINKEL, B. : The non i. i. d. strong law of large numbers in 2-uniiormly smooth Banach spaces. Probability Theory on Vector Spaces III - Lublin 1983 Lecture Notes in Math 1080, 90-118 [11 J HEINKEL, B. : Une extension de la loi des grands nombres de Prohorov. Z. Wahrscheinlichkeitstheorie verw. Gebiete 67 (1984), 349-362 [12J HEINKEL, B. : On Brunk's law of large numbers in some type 2 spaces. to appear in "Probability in Banach spaces 5" - Medford 1984 - , Lecture Notes in Math [13J HOFFMANN-J~RGENSEN, J. : Sums of independent Banach space valued random variables. Studia Math 52 (1974). 159-186 [14J HOFFMANN-J~RGENSEN, J. : Probability in Banach space. Ecole d'ete de Probabilites de St Flour 6 _ 1976 - Lecture Notes in Math 598, 1-186 [15J HOFFMANN-J~RGENSEN, J. and PISIER, G. : The law of large numbers and the central limit theorem in Banach spaces. Ann. Prob. 4 (1976), 587-599 [l6J KUELBS, J. : Kolmogorov law of the iterated logarithm for Banach space valued random variables. Ill. J. of Math. 21 (1977), 784-800 [17J KUELBS, J. and ZINN, J. : Some stability results for vector valued random variables. Ann. Prob. 7 (1979), 75-84 [18J

LEDOUX, M. : Sur une inegalite de H. P. Rosenthal et Ie theoreme limite central dans les espaces de Banach. Israel J. of Math. 50 (1985),290-318

[19 J

LEDOUX, M. : La loi du logarithme itere dans les espaces de Banach uniiormement convexes. C. R. Acad. Sc. Paris 300, Ser. I, n° 17 (1985), 613-616

[20J STOUT, [21 J

w.

F. : Almost sure convergence. Academic Press, New York 1974

YURINSKII, V. V. : Exponential bounds for large deviations. Theor. Prob. Appl. 19 (1974). 154-155

ON

THE

SMALL

BALLS

CONDITION IN

IN

THE

UNIFORMLY

CENTRAL

LIMIT

CONVEX

SPACES

THEOREM

Michel Ledoux Departement de Mathematique, Universite Louis-Pasteur 7, rue Rene-Descartes, F-67084 Strasbourg, France

Introduction. Let

E

be a Banach space. By random variable with values in

we mean a measurable map

X

with its Borel a-field

~(E)

probability measure on

~(E)

copies of

and, for each

X

with values in

E

measure on

such that the image of JP Denote by n

:2:

1 ,

Sn

(X) nnE:JN ~

(O,d, JP) by

X

into

E

IN

E

equipped

defines a Radon

a sequence of independent

Xl + .•• + X • A random variable n

is said to satisfy the central limit theorem (CLT

Ill) n (sn / r"

if the sequence

from some probability space

E

X

in short)

converges weakly to a Gaussian Radon probability

E.

In his remarkable work on the Glivenko-Cantelli problem, observed the following characterization of the

CLT

M.

[T]

Talagrand

for random variables with a

strong second moment which relates the central limit property to conditions on small balls: if the

CLT

lim

X

takes its values in

iff, for each

IlsJ

1m JP[ -

<

E

and

IE[llxI1 2 }

<

QQ

,

then

X

satisfies

E > 0 ,

E }

Iii (Actually, as detail led in

> o •

[T] ,

this equivalence holds in the more general setting

of non-separable range spaces and in the framework of empirical processes.)

Although it seems rather difficult to verify these conditions on small balls, the preceding property is intriguing since it reduces a central limit property in Banach spaces to some kind of weak convergence on the line by taking norm. This property also lies at some intermediate stage since, as we will see below, a random

45 variable K in

X with values in

E

satisfies the

CLT

iff there exists a compact set

E such that S

lim inf IP [ ...E. E K} n- CO /ri

> 0

(Sn/jll)n E ~

and the sequence

Ilsnll lim inf IP [ - -

< M}

>

is stochastically bounded as soon as for some

M> 0

0 •

v£

n-oo

M. Talagrand (oral communication) raised the question whether the equivalence he proved holds without the strong second moment assumption which is not necessary in general for the

CLT • In this note, we answer this question in a positive way

in uniformlY convex spaces. Precisely, we will establish the following result

THEOREM 1 • Let with values in

E

E. Then 2

(1)

lim t t-ao

(ii)

for each

be

a uniformly convex Banach space and X satisfies the

JP[ Ilxll > t}

CLT

X a random variable

iff

0

and E

> 0 ,

Ils)1 lim inf I P [ n-'"

vn

<

E

}

>

o .

This result will follow easily from a new quadratic estimate of sums of independent random variables in uniformly convex spaces obtained in

[L2] •

Preliminary results. We begin this section by a characterization of the which follows easily from the concentration's inequality of M. Kanter

CLT

[K] • I am

grateful to Prof. X. Fernique for useful informations on this result.

PROPOS IT ION 2 Then

X

. Let

satisfies the

lim inf IP[ n- OO

S n

vn

X

CLT

E K }

be a random variable with values in a Banach space iff there is a compact set

>

0

K in

E

E such t fl-'lt

(1)

.

46

Further, the sequence IlsJ lim inf JP[-

(Sn / [ii)n EC:IN

<

M }

is stochastically bounded iff for some

M> 0

o •

>

hi (1)

Proof. The necessity of

(2)

and

(S / Ill) :IN. Assume to begin wit h that nnE

implies the stochastic boundedness of X

is symmetric.

There exist

6 > 0

(2)

is obvious. Let us first show why

and

k

such that for all integers

o

k

~

k

0

n

and

Ilsnkll

IP[-- <

~

M }

>

6 •

>

MJk })"""2

By Kanter's inequality Ilsnll

3

( 1 + kJP[ - -

2

1

~

6

hi

and thus

-

It follows that the sequence JE[llxII

CI }

copy of of that

<

for all

CD

CI

1

'n) n E :IN (s n / V"

is stochastically bounded and also tl'Bt

< 2 • In the non-symmetric case, let

X; the symmetric random variable

X - X'

satisfies

M) and hence the preceding conclusions apply to JE(llxll} <

implies that

00

;

X'

(2 )

be an independent (with

2M

instead

X- X' • In particular, we have

therefore the strong law of large numbers combined with

(2)

X must be centered. Hence the conclusion to the second part of Pro po-

sition 2 holds by classical considerations involving Jensen's inequality. The first part is established in the same way.

Since in cotype 2 spaces, random variables stochastically bounded satisfy the

CLT

X such that

(Sn/jn)n E l'J

is

[p-z] , the previous proposition yields

immediately the following corollary.

COROLLARY 3 with values in

Let

E

E. Then

be a Banach space of cotype 2 and X satisfies the

CLT

iff

(2)

X

a random variable

holds.

47

We now turn to the small balls condition. Since for a centered Gaussian Radon probability, each ball centered at the origin of positive radius has positive mass, it is clearly necessary for a random variable for each

£

IlsJ lim inf JP[-

> 0 ,

>

}

2

showed that when JE[llxI1 } <

[T]

to satisfy the

£

CLT

that

0 .

Jri

n .... oo

M. Talagrand

<

X to satisfy the

(3)

CD,

is also sufficient for

X

CLT • For the sake of completeness, we reproduce here Talagrand's

proof of this result; it will illustrate the idea we will use next in uniformly convex spaces.

THEOREM 4 • Let 2 JE[llxI1 } <

that

Proof. Let

X be a random variable with values in a Banach space Then

00

•

£

> 0

X satisfies the

be fixed

<

liminf

£

}

and

CLT

6(£) > 0

6

(3)

iff

E such

holds.

be such that

6 •

>

Choose a finite dimensional subspace

of

H

E such that if

T

denotes the quotient

E.... E/ and 11.11 the quotient norm given by Ilr(x)11 = d(x,H) , then H 2 JE[IIT(X)11 } :s; 6.£2 • For each n, IIT(S )11 - JE[IIT(S )II} can be written as a

map

n

n

martingale n

IIT(S n )11 -

JE[l/T(S n )II}

with increments

d. , i ~

=

2: d. i= 1 ~

= 1, ••• ,n , such that, for each

i

[yJ) and thus by Chebyschev's inequality

(cf

S

F [IIIT(

Since

~ )11 -

;n

S

E[IIT( ~ )II} I

>

£

Jll

IITII:s; 1 , S

lim 1nf n ....

OO

JP[ IIT( ~)/1 In

<

£

}

>

6 ,

}

:s;

6 •

48 and hence, intersecting, S

lim sup JE[IIT(

< 2

n )II}

£

n .... oo

X therefore satisfies the

CLT

by classical arguments (cf

[P2]).

A short analysis of this proof shows the central role of the martingale trick 2 leading to the quadraUc estimate and of the integrability condition JE[llxI1 } <

CD

providing tightness at some point. An improved version, in uniformly convex spaces, of the previous martingale argument of Yurinskii was recently obtained

in

[L2]

in some work on the law of the iterated logarithm. It will allow us to establish Theorem

which thus characterizes in those spaces the

condition

(3)

and a moment condition which is necessary for the

Recall that a Banach space a

= 6(£)

6

CLT through the small balls

> 0

E

such that for all

CLT .

is uniformly convex if for each x,y

in

E with

Ilxll

= Ilyll

= 1

£

> 0 and

there is Ilx-yll:2: £

1 - II~II > 6 • According to a well-known fundamental result of G. Fisier

one has

2

[P1] , every uniformly convex Banach space admits an equivalent norm (denoted again

E

is p-smooth for some

p > 1

i.e.

11.11) with corresponding modulus of

smoothness

pet) satisfying

sup [ Hllx + tyll + Ilx - tyll) - 1, p(t),s; Kt P

for all

t > 0

1 }

Ilxll

and some positive finite constant

K.

This p-smooth norm is uniformly Frechet-differentiable away from the origin with derivative F (0)

and C

>

P

0

(cf

D : E - [a}

=

0 ,

then

....

E*

IIF (x)11 p

such that i f

F (x) p

= Ilx11 P- 1 for all

x

p 1 Ilxll - D(x/llxll)

=

in

for

x

I

0

and, for some constant

E

[H-J] ),

IIF (x) - F (y)11 P P

,;;

p 1 C II x _ y II -

for all

x,y

in

E.

The following lemma was the key point in the proof of the main result of It will allow to aChieve our wish in the next section.

(4)

[L2].

49

LEMMA 5 • Let satisfying

E

(4)

p > 1

be a p-smooth Banach space for some

Let also

(Y)i"n

E-valued random variables and let

with norm

11.11

be a finite sequence of independent bounded

S

written as a martingale

with increments

where

C

d

i

i

= l, ... ,n

such that, for each

is the constant appearing in

i

(4) •

Before turning to the proof of Thearem 1 , let us point out that a quotient E

norm of

holds true for any quotient norm of

E, property

constant

(4)

is also p-smooth, and, if

11.11

of a p-smooth Banach space

11.11

with uniform

C.

Proof of Theorem 1 • We may and do assume that

11.11

denotes the p-smooth

for some

p

> 1 for which

(4)

E

is equipped with a p-smooth

and Lemma 5 hold. By the previous remark,

these will also hold for every quotient norm with uniform constant moreover

p

C. We assume

<2

Condition

(i)

is well-known to be necessary for

X

to satisfy the

CLT

( [A-A-G] , [p-Z] ). Let us show the suffiency part of the theorem and assume first that

X

is symmetric. Proposition 2 and

(ii)

imply that the sequence

is stochastically bounded and thus, from the integrability results of

(Sn / /ll)n E:N [P2], we

know that

<

sup

CD

•

n

Let

E ~

0

be fixed. For each

n, define

(5 )

50

u.

1.

= u.(n) 1.

1, •• . ,n

i

n

U = L

and set number

(i)

u

i=l

n

and

combine to imply the existence or a real

(ii)

i

such that

0=0(£»0

< £}

lim im IP[ Ilun II

(s n / v" Tn) n

Since the sequence

°.

>

(6)

is stochastically bounded under

E}II

does not contain an isomorphic copy or

c

,Theorem 5.1 or

o

[p-zJ

is pregaussian, that is, there exists a Gaussian random variable the same covariance structure as

and

E

ensures that G in

E with

X. The integrability or Gaussian random vectors

H or

allows then to choose a rinite dimensional subspace denotes the quotient map

(ii)

E such that ir

T

E ..... Ej H '

We now apply Lemma 5 to the sum

T(U )

F

n

P

the Frechet derivative or the quotient norm or

will thererore denote below

EjH' For each

n, we have by

orthogonality,

n S

2p2

2 JE[F (T(U - Ui))(T(U.))}

L i=l

s

P

n

2C

+

2

1

~

E[llu _ u.11 2 (p-1)}

i=l

n

2 IE[IIT(U.)11 p}

r i=l

1.

2p2 n- JEtlIT(G)11 2 }

n

+

1.

(8 )

1.

2C 2 n JE[llu (n)11 2P } 1

since by independence 2 JE[F (T (U - u. ) ) (T (u. ) )} p

n

1.

1.

s

(where the supremum runs over the unit ball

(E/H)~

= Now, by symmetry and

(5) , ror each

i

1, •• • ,n

or the dual or

E/H) and

X

51

K

Further,

lit

,;;

S

JP[

lP[

Ilxll

n l-p

Ilxll ;:.

t } d t

2p

o

1

,;;

S

n

o

>

t

hi. } d

t

2p

so that

o

lim

by

(i)

and dominated convergence. These observations and

(8)

therefore imply that

lim sup n~OO

(by

(7)) and since

JE[IIT(U )II P }

lim sup n-

(6)

holds, by intersection,

<

E

P

n

CtO

which easily implies that

X

In the general case, let random variable so does

2

X - X'

satisfies the

X'

satisfies

CLT

(using

(i)

be an independent copy of (i)

and

(ii)

one more time).

X

; the symmetric

and thus the

CLT

and therefore

X.

Conclusion. It is an open problem to know whether Theorem 1 holds true in any Banach space; since we used the fact that random variables satisfying pregaussian in spaces which do not contain an isomorphic copy of

(ii)

are

co' a general

statement should probably include the condition

(iii)

X

is pregaussian

as an additional (necessary) assumption. It would also be interesting to know in what spaces, random variables satisfying CLT

iff the sequence

(sn / yll In) n Em

(i)

(and possibly

(iii)) verify the

is stochastically bounded (which is weaker

52

than

(ii) ). At the present, only trivial situations in which this happens (such

that cotype 2 spaces or spaces satisfying A-Ros(2)

[Ll]

like

L

P

(1,s; p < DO)

spaces) have been described.

References.

[A-A-G]

de Acosta, A., Araujo, A., Gine, E. : On Poisson measures, Gaussian measures, and the central limit theorem in Banach spaces. Advances in Prob., vol. 4, 1-68, Dekker, New York (1978). Hoffmann-Jprgensen, J. : On the modulus of smoothness and the G -conditions O!

in B-spaces. Aarhus Preprint Series 1974-75, n02 (1975). [K]

Kanter, M. : Probability inequalities for convex sets. J. Multivariate Anal. 6, 222-236

[Ll]

(1978)

Ledoux, M. : Sur une incgalitc de H.P. Rosenthal et Ie theoreme limite central dans les espaces de Banach. Israel J. Math. 50, 290-318 (1985).

[L2]

Ledoux, M. : The law of the iterated logarithm in uniformly convex Banach spaces. Trans. Amer. Math. Soc. (May 1986).

[Pl]

Pisier, G. : Martingales with values in uniformly convex spaces. Israel J. Math. 20, 326-350 (1975).

[P2]

Pisier, G. : Le theoreme de la limite centrale et la loi du logarithme itere dans les espaces de Banach. Scminaire Maurey-Schwartz 1975-76, exposes 3 et 4, Ecole Polytechnique, Paris (1976). Pisier, G., Zinn, J. : On the limit theorems for random variables with values in the spaces

[T]

L

P

(2 ,,; p < (0) • Zeit. fur Wahr. 41, 289-304 (1978).

Talagrand, M. : The Glivenko-Cantelli problem. Annals of Math., to appear (1984).

[y]

Yurinskii, V.V. : Exponential bounds for large deviations. Theor. Probability AppL 19, 154-155 (1974).

OF

SOME

REMARKS

GAUSSIAN

AND

ON

THE

UNIFORM

RADEMACHER

CONVERGENCE

FOURIER

QUADRATIC

FORMS

.lE-

M. Ledoux

M.B. Marcus

Departement de Mathematique

Department of Mathematics

Universi te Louis-Pasteur

Texas A & M University

67084 Strasbourg,

College Station, Texas 77843, U.S.A.

Fr~lce

1. Introduction

Let

{g}CD n n=O

be an L i.d. sequence of normal random variables with mean

zero and variance 1 and let

t

n n=O

of i.i.d. random variables where b e a sequence

be a Rademacher sequence, i.e. a sequence

fE}CD

P(E

O

= 1)

complex numbers satisfying

0f

Let

"Iam,n 12 ~

<00

{a

}

CD

m,n m,n=O

We define Gaussian

m,n and Rademacher Fourier quadratic forms as follows

(1. 1)

X (s,t) g

I:

m
am,ngmgn e

i(ms+nt)

(s,t) E [0,2TI]2

i(ms+nt)

(s , t) E [0, 2TI

and (1 .2)

a

I:

XE(s,t)

m
E E e

m,n m n

i

We are concerned with the uniform convergence a.s. of the stochastic processes

X (s,t)

and

N

n-1

I:

I:

g

lim N

~CD

xE(s,t) . (By convergence we mean

n=l m=O

and similarly for

XE(s,t).) It is known that if these series converge in

probability then they converge uniformly a.s. and in

*

LP

of their sup-norm. This

Professor Marcus was a visiting professor at the University of Strasbourg

when this work was initiated. He is very grateful for the hospitality and friendship that was extended to him while he was there. His work on this paper was also supported by a grant from the National Science Foundation of the U.S.A •.

54

is given in

[2J

forms (cf. also

[g I}

Let

n

and

[3J

respectively for Gaussian and Rademacher quadratic

[9J). and

[EI} n

be independent copies of

from the decoupling inequalities that for all

p

Ellx g (s,t)II

P

E[J

~

I:

m
n

~

a

as in (1.1) and (1.2) with finitely many non zero

(1 .3)

[E } . It follows

m,n

' i(ms + nt )II P am,ngmgne

and (1.4)

EIIx (s,t)/I E

P

I:

Ell

a

m
E E ' e i(ms+nt)II P

m,n m n

where the constants of equivalence only depend on

p

and

11.11

indicates

sup 2 I· I . We will show how (1.3) and (1.4) follow from the usual (s,t)E [O,2nJ decoupling inequalities in the Appendix.

We associate with

[xg(s,t)}(s,t) E[o,2nJ2

and

[xE(s,t)}(s,t)E [O,2nJ2

the

pseudometric (1. 5)

where

I: Ia 12 1e i (ms + nt) _ e i (ms ' + nt' ) 12 ) 1/2 m< n m,n

(s,t), (s',t') E [o,2nJ 2 . As usual let

entropy of radius

(

d( (s , t ) , (s ' , t ,) )

d

[O,2nJ2

E> 0

with respect to

in the pseudometric

N([O,2nJ2,d;E)

denote the metric

d, i.e. the minimum number of open balls of d

that covers

[O,2nJ2. A number of people

have recognized that

SCD logN([O,2 n J 2 ,d;E)

(1 .6)

dE

<

00

o

is a sufficient condition for the uniform convergence a.s. of (1.1) or (1.2). It appears for example in

[4J

(stated in a somewhat stronger form involving

majorizing measures). In fact C. Borell pointed this out to us and raised the question, can this sufficient condition be improved? The suspicion that (1.6) can be improved is reasonable because there are some obvious cases in which

55

xg (s,t)

is sufficient for the uniform convergence a.s. of these

is when

a

or

x (s,t) • One of E

is a product; (details will be given in § 2). Nevertheless,

m,n

no smaller function of the metric entropy than the one appearing in (1.6) suffices as a sufficient Condition for the uniform convergence a.s. of To be more precise, for all functions increasing and

lim x

~

= 0

f(x)/x

f : R+ ~ R+

such that

Xg(s,t)

or

f(O) = 0

XE(s,t) f

is

(and also satisfies a weak smoothness condition

CD

which will be given in § 2), we give examples of Gaussian and Rademacher Fourier quadratic forms which are not uniformly convergent a.s. but for which 2

CD

S

f(logN([0,2 n J ,d;E)) dE

<

CD

•

o

These examples also show that contrary to our experience with Gaussian processes no condition on the metric

d

alone can give necessary and sufficient conditions

for the uniform convergence a.s. of (1.1).

Pertaining to our study, X. Fernique

[6J

has obtained an important

corollary of the deep recent result of M. Talagrand

[13J

on the existence of a

majorizing measure for bounded Gaussian processes. Fernique's corollary deals with stationary vector valued Gaussian processes ; using this corollary (in a form adapted to our needs, Theorem 3.1 of this paper) and Theorem 1.1 of

[S,Chapter IJ

we obtain the following theorem which extends the equivalence between Gaussian and Rademacher Fourier series to quadratic forms.

Theorem 1.1. Let

tx/s,t)}(s,t)E[0,2nJ2

and

(x/s,t)}(s,t)E[0,2 n J2

as given in (1.1) and (1.2). If only finitely many of the coefficients non zero, then for all

(1 . S)

~

m,n

are

1

Ellx g (s,t)II P

where, as above, only on

p

a

be

11.11 =

sup 2 (s,t) E [0,2nJ

p . In particular,

and (l.S) holds in this case.

xg(s,t)

1.1

and the constants of equivalence depend

converges uniformly a.s. iff

xE(s,t)

does

56 Developping further the arguments of the proof of Theorem 1.1 , we characterize

xy (s,t)

the a.s. uniform convergence of

in terms of one-dimensional

entropies following Fernique's characterization of boundedness and continuity of vector valued stationary Gaussian processes. Assume for notational convenience that a

o

m,n

m~ n

if

(1 .9)

and set for

s, s', t, t'

d((s,O),(s' ,0))

and for

j

E

and

[0,2nJ d((O,t),(O,t'))

d (t,t') 2

1, 2

S

00

(1.10)

o

Recall that the space

C

(log N([0,2n J,d.;E))

1/2

dE.

J

, of a.s. continuous random Fourier series, is defined

a.s.

as the space of all sequences of complex numbers

a

=

[an}:o

in

-t 2

such that

t E [0,2nJ converges uniformly a.s. equipped with the norm

[a] The dual space

c*a.s.

space of multipliers

of

C

a.s.

has been carefully described by G. Pisier as a

M(2'~2) ; we refer the reader to

[10J, [8J

for further

details. We will not be directly concerned with this here although various interpretations of what we obtain can be given in the language of Define for each

m and

the one-dimensional sequences

and

(1.11) Set further for each

n

[lOJ.

T

in

C*

a.s.

and

j

, 2

t, t' E [0,2nJ

whenever it is defined. As a corollary of the results of Talagrand and Fernique we obtain ;

57

Theorem 1.2. Let

f

1.

Xg (S , t)} ( s , t ) E [ (I, 2n J2

finitely many non zero coefficients

E

a

m,n

Ilxg (s,t)11

Then for

+

j

= 1 or 2

J 1/ ([0,2n J,d 1 ) 2

+

(1.12)

be as given in (1.1) with only

+ T

sup J1/r([0,2nJ,d.) T E C* ~ J a.s.

IITII

$1

is

where the constants of equivalence are numerical constants and defined as in

(1.10) with

d~

instead of

J

d. J

We note that by considering Cauchy sequences Theorem 1.2 gives necessary and sufficient conditions for uniform also of

converge~ce

a.s. of

xg (s,t)

and, by Theorem 1.1 ,

XE(s,t) . Actually it is easily seen that, by Theorem 1.1 of

[S,Chapterr J ,

(1.12) also implies (l.S) .

It is customary to leave out the "diagonal" (Le. the terms

a

m,m

) when

studying random quadratics forms. For one thing (1.3) and (1.4) are no longer true, in general, if

a

m,m

J

0 . Furthermore, tte diagonal terms can be handled separately.

We write (1.13)

" 2im(s+t) "" a g e m m,m m

" im(s+ t) "" am,m e

+

m

By standard symmetrization argument and Theorem 1.1

[S,ChapterrJ , the second

series on the right in (1.13) converges u~iformly a.s. if and only if " imu "" a g e m m,m m

u E [0,2nJ

converges uniformly a.s .. The first series on the right in (1.13) is a deterministic Fourier series and the dichotomy betweel. unboundedness and uniform convergence a. s. is no longer relevant. Obviously in the Rademacher case one only gets on the right in (1.13)

" im(s+t) "" a e m m,m

58

2. Random Fourier quadratic forms and entropy

The sufficient conditions that we know for the uniform convergence a.s. of (1.1) and (1.2) follow immediately from well known results and techniques. We will present them here for the convenience of the reader.

Theorem 2.1. If

<

J1([o,2nJ2,d)

the Fourier quadratic forms

00

(1.1) and

(1.2) converge uniformly a.s. and

(2. 1)

where

2 ) 1/2

+

is a numerical constant. (2.1) is also valid with

x

E

C

C ( (I:

:s;

sup 2 Ix (s,t) I (s,t)E[O,2TIJ g

I

m< n

a

1

m,n

Conversely, if either of the Fourier quadratic forms J1/2([O,2nJ2,d) <

a.s. then

CD

easily obtain from the results of C > 0

P (

or

X

E

X

E

converge uniformly

•

be complex numbers satisfying

Proof. Let

constant

X g

replaced by

g

[12J

or

[14J

I: Ib 1 m
2

<

00

•

One can

that there exists an absolute

such that

I: b g g I m
> A ( I: Ib m
An estimate similar to (2.2) holds when

1

m,n

2 ) 1/2 )

[g} n

C exp ( - A/ C) ,

is replaced by

[E n } ,

'if A > 0 •

(see e.g. [lJ ).

Using these estimates Theorem 2.1 follows from the usual extensions of Dudley's theorem as presented for example in [11J

E exp ~ I

or

[5J

since we have that

x (s,t) - x (s',t') g

g

<

00

d((s,t),(s',t')) for some

~

> 0

and similarly with

x

g

replaced by

X

E

For the converse we note that by (1.3) the uniform convergence a.s. of (1.1) implies that of I: ( I: a g') g e n m,n n m m

ims

s E [o,2nJ

59

in which, to simplify the notation, we take independent from

am,n

=

°

if

m

~

n • Since

[g } is m

and is symmetric, the uniform convergence a.s. of

(1.1) implies, by Theorem 1.1

[8 ,Chapter I] , that of

s E [O,2n] ,

which is equivalent to

where

d

is defined in (1.9). Sinilarly we sec that

1

By the triangle inequality the metric

d

d((s,s'),(t,t'))

dpfinpo in (1.5) satisfies

+

Therefore (2.5)

and thus we see by (2.3), (2.4) ano (2.5) that the uniform convergence a.s. of (1.1) implies

J1//[O,2n]2,d)

<

CD

Exactly the same argument applies i f (1.1) is

•

replaced by (1.2) •

In an analogous fashion one can show that OO

S

r

~

(log N ([O,2n] ,d; E))

r/~

~dE

<

00

°

implies the uniform convergence a.~;. of r-dimensional tetrahedral (i.e. attention is restricted to the subset of

a

]TIl

7· • •

,m

for which

and Rademacher Fourier forms. Here, as above, distance in

L

2

m < m') < ••• < m ) Gaussian r 1 <-

r

d

denotes the corresponding

•

We now give examples which show that if one wishes to give metric entropy conditions for the uniform convergence a.s. of (1.1) or (1.2) solely in terms of the metric

d, then the sufficient: condition of Theorem 2.1 can not be improved.

60 1'(0) lim

f(x)

=

=

m . Furthermore we assume that for some C > 0 and all n

(Such a condition will follow if for some positive {,inite numbers f(2x)

:s; K

N. > N J

f(x)

for all

[b .}

large enough

o

M and

K,

tN.} be a sequence of integers such that

Let

M .)

x:;'

=d let

+ ••• + N. 1 J-

1

lim x/f(x) x - co

0 , be an increasing function satisfYing

J

be a sequence of positive numbers. Their precise

J

00

I: b~ N < j=1 J J

values will be specified later but we assume already that

2

00

Define

00

xg (s,t)

(s,t) E [0,2n]2

I: b. I: j=1 J m,n E I(j)

m
=

tNJ.}

*

0 • Note that and

tb J.}

tn

I(j) -

j:;, 1 ,

I(j)

1

+ ••. + N _

j 1

:s;

n < N + .,. + N } j 1

We will show that for some appropriate choice of

Nj

=

: N

depending upon

l' , the process

X

g

defined by (2.7) does not

converge uniformly a.s. even though

J

00

2

f(logN([0,2 n ] ,d;E)) dE

<

00

o

where

d

is the metric associated with

as given in (1.5).

X

g

By independence and Jensen's inequality we see that for each g g e i(ms

I:

b.

J m,n E I(j) m
and, by Remark 1.3

E

II

+nt )11

sup

m n

j :s; J

[S,Chapter VI] , for each

b.

Ell

J

J:;' 1 gmgnei(ms+nt)11

I:

m,n E I(j) m
j

I:

m,n E I(j) m
~ (E

:;,

(Throughout this example

I

intl2 g e sup I: n t E [0,2n] n E I(j)

-

/2N.) J

C N. log F. J

J

C will denote a strictly positive constant, possibly

changing from line to line.) Thus we have that

61

sup

(2.8)

E

J I:

II

j=1

J ;;, 1

II

g g e i(ms +nt) b. I: m n J m,nE I(j)

;;,

C

sup j ;;, 1

b.N .log N. J J

J

m
d((s,s'),(t,t'))

2

00

I:

I

"

b. '" e j=1 J m,n E I(j)

(

i(ms +nt) _ ei(ms' +nt') 12 )1/2

m
where

6

6(s,s')

+

6(t,t')

denotes the one-dimensional metric given by

~ b~

(

6(t,t')

j=1

N.

le int _ e int ' 12 ) 1/2

I:

J n EI(j)

J

It follows that 00

~

2

S

f(2 log N ([O,2nJ,6;E)) dE

o s > 0

We now estimate this entropy. For

~(s )

sup

lu I ~s

and each

j

&. (s)

6(u,O)

sup ~

lui

J

[S, p.123-124 and Lemma 3.6 Chapter IIJ

Following

we define (I:

s nE I(j)

le inu _ 11 2 ) 1/2 •

we see that

00

S

o

f ( 2 log N ([ 0 , 2n J , 6 ; E)) dE 00

1

/

C ( ( I: b~ N~ ) 1/2 j=1 J J

~

+

S

o

00

C ( ( I: b~ N~ ) 1/2 j=1 J J

~

1

f ( 2 locr

00

+

b.,;'N":"

I:

j= 1

J

J

~ ) do (s) )

S1

1

f (2 log - ) do. (s) ) 0 s J

Furthermore,

So1 f(21og

1

- ) dO.(s) s J CD

~

Now for each

j

r; .(s) J

and

f(2)

&.(1) J

I:

+

k=1

s > 0 N .-"1 J

sup (I: lu I ~s n=O

Ie inu

+

&.(2- k ) (f(2k+2) - f(2k)) • J

62 Let us denote by

To(s)

TJ (s)

to see that

~ 2 ~ , however, when J

(

J

for

N.-l

~

'1". 2 -k) A

the first torm of the right hand side of (2.9). It is easy

J

2- k (

~

n2) 1/2

n=O

sufficiently large. Using these estimates for

j

'1".(2 -k) A

and (2.6) we see

J

that for

sufficiently large

j 00

L

k=l

[log2 Nj]

T.(2-

k

) (f(2k+2) - f(2k))

2 ~ (f(2k+2) - f(2k))

L

J

J

k=l 00

L

+

k=[ log N.J+ 1 2 J

c IN:

~

By a similar decomposition of the sum on 00

~ [(N 1+ ••• +N j _ 1 ) k=l J L

~

CD

L

f f ( N. 2-

k=l

J

2-

k

k

II

II

J

k

f ( 310g N . )

J

we get

2] (£(2k+2) - f(2k)) ~

2) (£'(2k+2) - f(2k))

c $

J

J

f (3log N . ) J

Putting this all together we see that

(2.10)

Since

+

f

satisfies

lim x

~

~ CD

J J

J

we can first choose

00

tNJ.}

such that

o

lim j

o ,

fex) / x

b.N.f (31ogN.) ) •

logN. J

and then take b. J

(j 2 N.f(3logN. ) )-1 J

J

j

For these Choices, we see from (2.10) that

~

1 •

Jf([O,2n]2,d)

<

CD

even though,

by (2.8) and the integrability properties of Gaussian quadratic forms, does not converge uniformly a.s ..

x (s,t) g

63 Thus (1.6) Can not be replaced by (1.7), with any increasing function satisfying

f(x) /x = 0

lim

f

and (2.6), as a sufficient condition for the

X~CD

uniform convergence a.s. of (1.1). Note that exactly the same argument applies for the corresponding Rademacher Fourier quadratic form, i.e., with

In some cases

<

J1/2([0,2nJ2,d)

tE E } m n

replacing

is necessary and sufficient for the

00

uniform convergence a.s. of (1.1) ilnd (1.2). One of these, which is trivial, is when the coefficients

fa m,n }

vanish outside some one-dimensional set of indices.

1.

We write (1.1) in the form

(2.11)

(s,t) E [0,2nJ2

2: k

m < n

where

are non-negative integers. Using Gaussian decoupling we see that this

series converges uniformlY a.s. if and only if (s,t) E [o,2nJ

(2.12) converges uniformly a.s. where tgmkgrik}

f

g

1. mk

2

gnk }

Theorem 2.2. Let

o

E

[0,2nJ

x (s,t ) g

and

CD

it follows from Theorem 1.1

Once again exactly the same argument

•

in (2.11) is replaced by 2

J1/2([0,2nJ ,d)

marginal processes formed from

t

<

J1/2([0,2nJ ,d)

The finiteness of

the processes

k

that the series in (2.12) and consequently (2.11) converges uniformly

a.s. if and only if holdS if

is an independent copy of

tg' } nk

is an independent symmetric sequence in

[8 ,Chapter I J

2

0

s

o

E

uniformly a.s. for some

Xg(s,t)

x g (s,t) and

tEmkEnk}

also implies the continuity a.s. of the and

xE(s,t) •

be as given in (1.1). Then if

X (s ,t) g

0

are uniformly convergent a.s. for each fixed

[0,2nJ • Conversely if t

o

E

This theorem is also valid if

[0,2nJ X

g

(o,t)

and

s

o

X (s,t ) g

E

0

[0,2nJ

is replaced by

and then

X (s ,t) g

0

converge

J 1/2 ([0 ,2n J2 ,d)

xE(s,t) •

<

00

•

64 Proof. Let

(2.13)

d (s,s') 1 ~

N([0,2TI],d.;2E) J

We will show this for (sk,t ) , k

j

and

=

d (t,t') 2

N([0,2 TI ],d;E)

1 . The proof when

k = 1 , ... , N([0,2TIJ2,d;E)

that cover

[0,2TI]2

Let

be as defined in (1.9) . Let us note that

in the metric

j

j

=

1 , 2 •

2

be the centers of the balls of radius

d

For a fixed

be some fixed element in

is completely similar. Let

k

E

consider

B . It follows from the triangle inequality E,k

that B E,k

C

U £(s,O) E [0,2TI]2

since

d1(s'~k)

Since that i f

=

d((S'O)'(~k'O))

J 1//[0 ,2 TI J2 , d)

<

we have verified (2.13) when

then

CD

g e m

J 1//[0, 2TI], d )

1

ims

<

j = 1 • It follows

and this implies that

CD

s E [0,2 TI J

converges uniformly a.s .• By Theorem 1.1

[8 ,Chapter I J , the uniform convergence

a.s. of (2.14) is equivalent to that of s E [0, 2TI J

which, by decoupling inequalities similar to (1.3), implies the uniform convergence a.s. of m~ n

X (s,t ) . (To avoid confusing notation assume that g

a

0

in all these series.) A similar argument shows that

m,n

X (s ,t) g

0

o

for all

converges

uniformly a.s •• The converse follows from the proof of Theorem 2.1 since in the relevant part of the proof of Theorem 2.1 the only property of the continuity of X (s,t) g

that is used is that its marginals are continuous. All the above

statements remain valid when

xg(s,t)

is replaced by

The follOWing corollary is immediate.

xE(s,t) .

65 Corollary 2.3. Assume that

12

• Then

<

J 1/2 ([0 ,2n J2 , d)

convergence a.s. of

Xg(s,t)

2 1.f a }

Proof. Since

m

00

and

a

m,n

=

a a

[an } is a real sequence in

where

m n

is necessary and sufficient for the uniform x£(s,t) .

E .{., ,,1 and because of I:

uniformly a.s. if and only if

amgme

ims

x (s,t) g

converges

converges uniformly a.s •. The result

m

now follows from Theorem 2.2. The same proof

is valid for

X£ (s, t) •

Now let us consider stochastic processes of the form •

I: a ~ g g e m,n m,n m n m< n

where

[am,n}

and

[gn}

i(ms+nt)

(s,t) E [0,2nJ

P (£0,0

[g} • It follows from Theorem 1.1 n

the series in (2.15) converges uniformly a.s. if and only if where Let

d

is a doubly

are as given in (1.1) but where

indexed i.i.d. sequence of random variables with which is independent of

2

P ( £

1

= -1 )

0,0

"2

,

that

[8 ,Chapter I J

J1/2([0,2nJ2,d) <

00

is as given in (1.5). This observation has an interesting interpretation.

(O,J,p)

be a probability space on which

[£m,n}

w E 0

is defined. For each

we consider the Gaussian Fourier quadratic form ) I: a £ ( wgge m,n m,n m n m
(2.16)

The metric

d

of

Furthermore, i f

wE 0

i(ms +nt)

(s,t) E [0,2nJ2 .

(as defined in (1.5)) is the same for all these processes irregardless J1//[0,2nJ2,d) <

00

then for almost all

wE 0

the

series in (2.16) converge uniformly a.s. (With respect to the probability space supporting

[gn})' However, the series in (2.16) do not necessarily converge

uniformlY a.s. for all

w EO. This can be seen from the examples we just gave of

Gaussian Fourier quadratic forms which do not converge uniformly a.s. but for which 2 J / ([0,2nJ ,d) 1 2

<

00

•

Thus, unlike the random Fourier series studied in

continuity of Gaussian Fourier quadratic forms is not determined by the d

given in (1.5). Once again exactly the same argument applies if

replaced by

[8J 2 L

[gn}

[£ n } in (2.16).

Nevertheless, entropy still can be used to characterize a.S. uniform

is

metric

66 convergence of Gaussian and Rademacher Fourier quadratic forms. However, as we will see in the next section, it is necessary to consider the supremum of classical onedimensional entropies over a family of metrics.

3. Gaussian random Fourier series with coefficients in a Banach space

Fernique's corollary

Theorem 3.1. Let sequence of elements of

E t

[6J

(B,II.II)

of Talagrand's theorem

[13J

be a Banach space with dual

is the following.

B*. Let

[x} n

be a

B with only finitely many non zero terms. Then

sup II I: g x e int II n n E [0,2nJ n

E

I I: gn xn II n

+

sup E sup lI:g<x*,x>eintl Ilx*II':;l t E [0,2nJ n n n

x* E B* •

where

Therefore uniform convergence a.s. of random Fourier series with coefficients in a Banach space is characterized through conditions involving a family of classical one-dimensional entropies. We will see in the proof of Theorem 1.2 that (3.1) can be used to obtain a similar result for Gaussian and Rademacher Fourier quadratic forms. Using a set of one-dimensional entropy conditions instead of a single two-dimensional entropy condition to characterize uniform convergence a.s. of random Fourier quadratic forms seems to be necessary, as was shown by the class of examples described in § 2. Theorem 3.1 sheds some light on these examples. Indeed, the quadratic form (2.7) can almost be realized as a random Fourier series with coefficients in a Hilbert space. Define a sequence

[x } n

setting 'if J :;, 1 ,

where

[~}

'if n E I(j) ,

x

n

denotes the canonical basis of

x(t)

t

-t 2

E [0, 2n J

• Consider

of elements in

t2

by

67 Clearly I:

+

I: b.N. J J J

+

min

< x ,x > g g e m

Ilx(t)11

tb.}

and

J

2

i(m -n)t

m n

I: g g cos((m -n)t) m n J m,n E I(j)

2 I: b.

J

m

Thus

n


is closely related to the quadratic form (2.7). For the choices of in § 2,

tN.} J

x.*

(

=

J

X(t)

is unbounded a.s. since for

1 ) 1/2 I: e kE I(j) k Nj

( Ilx*11 J

1 )

we have sup I I: <x*,x>geinti t E [O,2n] n J n n

E

E

sup tE[O,2n]

I b:/2

C (b.N .1ogN . J J J

J

f

int

I

)1/2

C. Likewise the examples of § 2 show that no condition

for some absolute constant of the form

J

g e nEI(j) n I:

([O,2TI],6)

<

m

(see (1.7)) where

6(t,t') is sufficient for the uniform convergence a.s. of the Gaussian quadratic form in (3.2). (The relationship between Theorem 3.1 and metric entropy conditions is clearly E

sup II I: g x eintll t E[O,2n] n n n

where d At,t') x

n

We now use Theorem 3.1 to obtain Theorem 1.1. Proof of Theorem 1.1. We will give the proof in the case (1.4) we will prove this theorem with X'(s,t) g

Xg(s,t) and XE(s,t) ( s , t) E [

p ~ 1 • By (1.3) and

replaced respectively by

°,2n ] 2

68

and , i(ms+ nt) a E E e I: m,n m n m
X;(s,t)

where, to simplify the notation we take finitely many of the

and

m ;;, n

if

°

'"

m,n

are non zero. Let us define and

[g'} n

and

[O,2n]2

Recall that only

[gm}

and

[E } m

E

wE 0

E,E ,E, g E g

(0)<0' ,3<x3<' ,PXP')

denote expectation on the product space

and consider

f " g (w)e L ~ a m m,n m

ims

on the

on the probability space

[E '} n

and denote their corresponding expectation operators by

EE' • Let

us fix

m,n

(0,3< ,p)

probabili ty space

(0' ,J' ,p' )

a

a

E

(s,t)

}

. as a sequence In

in the Banach space of continuous complex valued functions on

n

Let

of elements

[O,2TI] • By Theorem

3.1 we have

E , sup g s,t

1 I:

( I: a

n

m

g (w) e ims) ,int e m,nm gn

+

By Theorem 1.4

[8 ,Chapter I]

E ,sup g s

1

sup sup s Eg , t

[g'}

we can replace

1

I: ( I: a g (w)e n m m,n m

[E'}

by

n

n

(3.3). Doing this and taking expectation with respect to

E sup IX'(s,t) I

s,t

g

E sup s

1

I: ( I: n m

f

L

imS

) g'eintl n

in the last term in

gm}

we get

ims am,ng~) gme

(3.4) +

E

g

ims E ' e int) g e sup I I: ( I: a sup E m,n n m E' t m n s

Let us denote the first and second term to the right of the equivalence sign in (3.4) by

I

and

I I . By Theorem 1.4

2 1 ) 1/2 E e E sup I I: ( I: la m,n m s m n

I

E sup s In analyzing

II

(3.5)

[8 ,Chapter I]

II :s;

I: ( I: a E' ) E e ims I m,n n m m n

we repeat the steps of

E sup s,t

ims

1

I: ( I: a m,n En' e int) g e m m n

E'e E sup \ I: ( I: a m,n n t n m

int

I

:s;

C3 . 3)

applied twice

and

E sup s,t

C3. 4 )

1

x;(s,t)

1

and obtain

ims

) gm l + EE t sup E sup 1I: (I: a E 'e int) g e ims t g s m n m,n n m

1

69 Denote the first term to the left of the equivalence sign in (3.5) by second by

I'

and the

II' • Using the same argument as above we have

E sup II: ( I: a g ) E'e m,n m n t n m

I'

I: ( I: a E) C'e m,n m n n m

E sup t We use Theorem 1.4

[S ,Chapter I]

on

int

int

s;

E sup I

s,t

II'

x~(s,t) I

and obtain

' int) E e ims EE' sup EE sup I I: ( I: a m,n En e m t s m n

II'

E sup I x~(s,t) I s,t

s;

Thus we have shown that E Ilx~(s,t)11

(3.6)

s;

C E Ilx~(s,t)11 C. The reverse inequality in (3.6) is obtained by

for some absolute constant

the contraction principle. This completes the proof of Theorem 1.1.

We finally prove Theorem 1.2.

Proof of Theorem 1.2. It will be sufficient to show that

E Ilx (s,t)11 g

+

E

sup E [0, 2TI]

t

sup

+

T E C*

a.s.

II for

j

=

1 or 2

n

m

E sup sE [0,2TI]

m

Til s; 1

where as before

a

m,n

= 0

if

m

~

n

have been defined in (1.11). Indeed, Theorem 1.4

tAj} m

and where the sequences [S,Chapter I]

then clearly

implies (1.12). Following the notation of the preceding proof we show (3.7) for j

=

2

and with the left hand side replaced by

Ellx'(s,t)11 g

proof of Theorem 1.1 (cL (3.3), (3.4) and the estimate of

by (1.3). As in the I), we have

70

E sup s,t

(3.8)

I X·g (s , t) I

E sup s

II

where

is the norm on

[.]

m

n

1

m,n

E sup E • sup g g t s

+

Let us call

I: ( I: la

2 )1/2

g eimsl m

( I: a g e imS) g' e m,n m n

I I: n

int

ITt

the second term to the right of (3.8). It is plain that

C . (3.7) then follows easily from a new application a.s.

of Theorem 3.1 but this time in

C . Indeed, a.s.

II

sup

+

T E C*

IITII and by Theorem 1.4

a.s. :s; 1

E sup s

[8 ,Chapter I] ,

2 E [I: Amgm ] g m

E sup

I: ( I: a

n

t

E sup t

m

I: ( I:

n

I

g) 9' e int

m,n m

n

lam,nI2)'/2g~CLnt I

m

Appendix. Proof of (1.3) and (1.4) : the argument that we give here was shown to us by Gilles Pisier. We will first prove (1.4). It follows from Theorem 2 , [7} for

that

p;;' 1

(A. 1 )

E II

I:

m
p . Therefore by the triangle

inequality , i(ms+nt)II P E E e CEil I: a p m,n m n m
C'

p

E II

I:

m
a

E E'ei(ms+nt)II P

m,n m n

E II

I:

m
a

m,n

,)i(ms+nt)II P • , ( EE+EEe m n

m n

71

Let

8 0

be a real valued random variable uniformly distributed on

[8n } be an i.i.d. sequence. Let

sn

=

in8 en,

s~ =

[8'} n

[O,2TIJ

be an independent copy of

and let

[8}. We define n

in8' e n . It follows from the contraction principle for quadratic

forms, Theorem 1 , [7J

that (A.2) holds i f Ell

(A. 3) We replace

L: a (S S'+S'S )ei(ms+nt)II P m
(s, t)

and

expression on the right in (A.3) where

s

o

and

t

o

by

(s-s ,t-t ) o

are fixed in

0

H .

in the

[0,2TIJ • These

changes do not change the numerical value of this term, i.e.

Finally we note that by Jensen's inequality 2TI 2TI

So S0 :;" Ell

H ds dt o 0

, L: a (S S +_1_ m< n m, n m n 4TI 2

S2TIS2TI e . [ ( n-m ) so+ ( m-n ) to J ds l

0

0

dt 0

0

Thus we have obtained (A.3) and consequently (1.4). The same argument works in the Gaussian case. We introduce in place of

[S)

and

[S'} n

in (A.3) where

as defined in the introduction and take Since

g

n

G'} n

and

G'} n

and

[g' } n

to be an independent copy of

Gn }

gn

=

gn + ig'n

for

Gn } [gn}

is rotationally invariant the exact same argument as above gives (1.3).

References [lJ

A. Bonami : Etude des coefficients de Fourier des fonctions de

LP(G) • Ann.

Inst. Fourier (Grenoble) 20 , p. 335-402 (1970). [2J

C. Borell ; Tail probabilities in Gauss space. Vector space measures and applications, Dublin 1978. Lecture Notes in Math. 644 , p. 71-82 , Springer (1979) •

72 [3J

C. Borell : On the integrability of Banach space valued Walsh polynomials. Seminaire de Probabilites XIII. Lecture Notes in Math. 721 , p. 1-3 , Springer (1979).

[4J

C. Borell: On polynomials chaos and integrability. Prob. and Math. Stat. 3 , p. 191-203 (1984).

[5J

X. Fernique : Regularite de fonctions aleatoires non gaussiennes. ECole d'ete de St-Flour 1981. Lecture Notes in Math. 976 , p. 1-74 , Springer (1983).

[6J

X. Fernique : Fonctions aleatoires gaussiennes

a

valeurs vectorielles.

Preprint (1985). [7J

S. Kwapien : Decoupling inequalities for polynomial chaos. Preprint (1985).

[8J

M.B. Marcus and G. Pisier : Random Fourier series with applications to harmonic analysis. Ann. Math. Studies 101 , Princeton Univ. Press (1981).

[9J

G. Pisier : Les inegalites de Khintchine-Kahane d'apres C. Borell. Seminaire sur la geometrie des espaces de Banach 1977-78, Ecole Polytechnique, Paris (1978).

[10J

G. Pisier : Sur l'espace de Banach des series de Fourier aleatoires presque surement continues. Scminairc sur la geometrie des espaces de Banach 1977-78, Ecole Polytechnique , Paris (1978).

[11J

G. Pisier : Some applications of the metric entropy condition to harmonic analysis. Banach spaces, harmonic analYsis and probability, Proceedings 1980-81. Lecture Notes in Math. 995 , p. 123-154 , Springer (1983).

[12J

M. Schreiber

Fermeture en probabilite des chaos de Wiener. C.R. Acad. Sci.

Paris, Serie A , 265 , p. 859-862 (1967). [13J

M. Talagrand : Regularite des processus gaussiens. C.R. Acad. Sci. Paris, Serie I , 301 , p. 379-381 (1985).

[14J

D. Varberg : Convergence of quadratic forms in independent random variables. Ann. Math. Statist. 37 , p. 567-576 (1966).

RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR EMPIRICAL PROCESSES by Pascal MASSART * universite Paris-Sud U.A. CNRS 743 "Statistique Appliquee" Mathematiques, Bat. 425 91405 ORSAY (France)

SUMMARY

In this paper we study the uniform behavior of the empirical brownian bridge over families of functions

F bounded by a function

pendent with common distribution

pl.

F (the observations are inde-

Under some suitable entropy conditions which

were already used by Kol~inskii and Pollard, we prove exponential inequalities in the uniformly bounded case where

F is a constant (the classical Kiefer's inequality

(1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where

F belongs to

L2+o(p)

improve on DUdley, Philipp's results (1983) whenever

with

°E ]0,1]

(our results v

F is a Vapnik-Cervonenkis

class in the uniformly bounded case and are new in the unbounded case).

1.

Introduction

2.

Entropy and measurability

3.

Exponential bounds for the empirical brownian bridge

4.

Exponential bounds for the brownian bridge

5.

Weak invariance principles with speeds of convergence

6.

Strong invariance principles with speeds of convergence

APPENDIX

1. Proof of the lemma 3.1. 2. The distribution of the supremum of a d-dimensional parameter brownian bridge 3. Making an exponential bound explicit.

Invariance principles, empirical processes, gaussian processes, exponential bounds.

74

1. I NTRODUCT ION 1.1. GENERALITIES. Let

(X,X,P)

be a probability space and

(x n )n>1

be some sequence of indepen-

dent and identically distributed random variables with law enough probability space

P, defined on a

rich

(0,A,Pr).

1 n Pn stands for the empirical measure L Ox. and we choose to call empirical n i~1 1 brownian bridge relating to P the centered and normalized process vn =!n (p n-pl. Our purpose is to study the behavior of the empirical brownian bridge uniformly over F, where F is some subset of L2(P).

More precisely, we hope to generalize and sometimes to improve some classical results about the empirical distribution functions on lR d (here F is the collection of quadrants on lR d ), in the way opened by Vapnik, Cervonenkis and Dudley. In particular, the problem is to get bounds for: (1.1.1)

Pr (

Ilv n II

F > t)

, for any positive

where 11.11 F stands for the uniform norm over

t,

F and to build strong uniform approxi-

mations of vn by some regular gaussian process indexed by convergence, say (b n )

F with some speed of

First let us recall the main known results about the subject in the classical case described above.

We only submit here a succinct bibliography in order to allow an easy comparison with our results (for a more complete bibliography see [26]) Concerning the real case

(d~1)

, the results mentioned below do not depend on

P and are optimal :

(1.1.1) is bounded by

C exp(-2t 2),

to Dvoretsky, Kiefer and Wolfowitz [24] (C

where ~ 4/2

C is a universal constant according according to [17]).

75 1.2.2.

The strong invariance principle holds with

~ , according to Komlos,

bn

In

Major and Tusnady [37] . In the multidimensional case (d

~

2).

1.2.3.

C(E:) exp (-(2-E:)t 2 ),

(1.1.1) is bounded by

Kiefer [34].

In this expression

for any

(:

> 0,

according to

[ cannot be removed (see [35] but al so [28]).

1.2.4.

The strong invariance principle holds with

b n

n- 2(2d-1) Log(n),

according

to Borisov [8]. This result is not known to be optimal, besides it can be improved when uniformly distributed on 1.2. 5.

If

d

=

[0,1]d

In this case we have :

2 :

(Log(n)) 2 , according to

The strong invariance principle holds with

In

Tusnady [50]. 1.2.c. If

d

P is

>3

:

1

The strong invariance principle holds with

b n

=

n-

3

2Td+TI (Log(n))2, according

to Csorgo and Revesz [14] 1.2.5 and 1.2.6 are not known to be optimal. Let us note that even the asymptotic distribution of [I\inll F is not well known (the case where

d = 2 and

P

is the uniform distribution on

[0,1]2

is studied

in [12]). Now we describe the way which has already been used to extend the above results.

v

Vapnik and Cervonenkis introduce in [51] some classes of sets - which are generally called V.C.-classes - for which they prove a strong Glivenko-Cantelli law of large numbers and an

~xponential

bOLnd for (1.1.1)

76

P. Assouad studies these classes in detail and gives many examples in [3] (see also [40] for a table of examples). The functional P-Donsker classes (that is to say those uniformly over which some central limit theorem holds) were introduced and characterized for the first time by Dudley in [20] and were studied by Dudley himself in [21] and later by Pollard in [44]. Some sufficient (and sometimes necessary, see [27] in case bounded) conditions for

F is uniformly

F to be a P-Donsker class used in these works are some kind

of entropy conditions Conditions where functions are approximated from above and below (bracketing, see [20]) are used in case

F is a P-Donsker class whenever

restricted set restricted set of laws on

X (p

P belongs to some

is often absolutely continuous with

respect to the Lebesgue measure in the applications) whereas Koltinskii and Pollard's conditions are used in case

F is a P-Donsker class whenever

P belongs to some set

of laws including any finite support law (the V.C.-classes are - under some measurability assumptions - the classes of sets of this kind, see [21]). In our study we are interested in the latter kind of the above classes. Let us recall the already existing results in this particular direction . Whenever

• F is some V.C.-class and under some measurabil ity conditions, we have:

1.3.1.

(1.1.1) is bounded by

C(F,d exp(-(2-dt 2 ) for any

E

in ]0,1], according to

Alexander in [1] and more precisely by : in [2] * where

D stands for the integer density of F (from Assouad's terminology in [3]) .

1.3.2.

2

( 1. 1.1) is bounded by

*

(~ )) exp(-2t 2) , according to Devroye in [16].

Our result of the same kind (inequality 3.3.1°)a) in the present work) seems to have been announced earlier (in [41]) than K. Alexander's one.

77

1.3.3.

The strong invariance principle holds with

b n

=n

2700(D+1) , according to

Dudley and Philipp in [23]. Now let us describe the scope of our work more precisely.

2. ENTROPY AND MEASUKABILITY. From now on we assume the existence of a non-negative measurable function such that

If I ~

F,

for any

f

F

in F.

v

We use in this work Kolcinskii's entropy notion following Pollard [44] and the same measurability condition as Dudley in [21] .

Let us define Kolcinskii's entropy

notion. Let and 2.1.

p be in

p~p)(X)

[1,+oo[

.

A(X)

stands for the set of laws with finite support

for the set of the laws making

FP integrable.

DEFINITIONS. Let £ be in

]0,1[

and

Q be in

p~p) (X) .

N~P)(£,F,Q) stands for the maximal caroinality of a subset G of F for which:

holds for any

f,g

in G with

f~g

(such a maximal cardinality family is called an

sup N~P~(.,F,Q) QEA(X) L09(N~P)(.,F)) is called the (p)-entropy function of (F,F).

£-net of

(F,F)

relating to Q).

We set

N~P)(.,F)

=

The finite or infinite quantities : inf (s>O e(P)(F) = inf (s>O F

limsup £SN~P)(£,F) < oo} £->-0 limsup £SL09(HF(P)(£,F)) < oo} £->-0

are respectively called (p)-entropy dimension and (p)-entropy exponent of

(F,F).

78

Entropy computations.

We can compute the entropy of

F from that of a uniformly bounded family as

follows

{~1(F>0)' fE n , then

Let I

N~P)(.,F) < NIp) (. ,I) For, given so

£

O(F P)

EA(X)

0 in A(X), either O(F)=O and so N~P)(.,F,O)

1, or

O(F) > 0,

and then

NF(P)( .,F, 0) -- N(P)( 1 .,I, Some other

properties of the (p)-entropy are collected in [40].

The main examples of uniformly bounded classes with finite (p)-entropy dimension or exponent are described below.

According to Dudley [20] on the one hand and to Assouad [3] on the other we have

, whenever S [3]).

is some V.C.-class with real density

d

(this notion can be found in

Concerning V.C.-classes of functions, an analogous computation and its appli-

cations are given in [45] • See also [21] for a converse.

Let d be an integer and

k

D

a

be some positive real number.

We write

S for the greatest integer strictly less than

Whenever

x belongs to

Rd and

for the differential operator

k

to

J1d ,

I kl

a.

stands for

d Ik I

k

k +· •• +k d

1

and

k

ax, 1., .ax dd Let 11.11 be some norm on Let

Rd •

a,d be the fami ly of the restrictions to the unit cube of s-differentiable functions f such that: f\

R

d of the

79

IDkf(x) I

max sup Ik 1~13 xERd

+

max su p.JDkf(x)-Dkf(y)1 - <1 Ik I=13 Xfy II x-y II a-13 -

Then, according to [36] on the one hand and using Dudley's arguments in [19] on the other, it is easy to see that: d

a

Measurability considerations.

• Durst and Dudley give in [21] an example of a V.C.-class S such that

So some measurability condition is needed to get any of the results we have in view. So from now on we assume the following measurability condition (which is due to Dudley [21]) to be fulfilled:

(M)

.(X,X)

is a Suslin space

There exists some auxiliary Suslin space from V onto F such that (x,y) ->- T(y) (x) and we say that

(V,Y)

and some mapping

T

is measurable on (XxX, X III Y )

F is image admissible Suslin via (V,T) •

This assumption is

esse~tially

used through one measurable selection theorem

which is due to Sion [47] (more about Suslin spaces is given in [13]).

2.4. THEOREM. ------Let on

X.

H be some measurable subset of Then

XxX.

We write

A for its projection

A is universally measurable and there exists a universally measu-

rable mapping from A to

V whose graph is included in

H.

A trajectory space for brownian bridges.

We set : l~(F) = {h:

We consider

l~(F)

F->-lR ; hoT

is bounded and measurable on

as a measurable space equipped with the

(V,Y)}. a-field generated

80

by the open balls relating to 11.11 a-field because

(which is generally distinct from the Borel F

l~(F)

is not separable).

This trajectory space does not depend on

P any more (as it was the case in

[20]) but only on the measurable representation

(Y,T)

of F .

From now on for convenience we set : /'-

( \I , A, Pr) "-

where

/'0

OO

(X°"x [0, 1],X

III

B([ 0, 1] ), P

00

III

Ie )

stands for the Lebesgue measure on [0,1], B([O,1]) for the Borel a-field /\-. /\,. oo on [0,1] and (Xoo,Xoo,p ) for the completed probability space of the countable prooo duct (Xoo,Xoo,p ) of copies of (X,X,p) • Ie

The following theorem points out how

l~(F)

is convenient as a trajectory

space.

n

For any a in

L i=1

a.a 1

Xi

is measurable from

\I

to

l~(F).

Moreover, setting Ub(F) = {h : F ~ R ; h is uniformly continuous and bounded on

(F,pp)}'

Ub(F)

is included in

l~(F).

Provided that

(F,pp)

is totally boun-

ded this inclusion is measurable.

L2(P)

Where 2

ap

is given the distance

Pp : (f,g)~ap(f-g), with For a proof of 2.5. see [21] (sec. 9) and [40] where it is

f~ p(f2)_(p(f))2

also shown that many reasonable families (in particular A d and the "geometrical" a,

V.C.-classes) fulfill (M) . 2.6. -----REMARK.

Since whenever

F fulfills (M) it follows from [21] (sec. 12) that IIPn-PII F-+-O a.s.

N~1)(. ,F) < sup

QEP~2) (X)

00

and therefore:

N~2) (E,F,Q) ~ N~2) (~,F) for any E in ]0,1 [ .

This implies that the local behavior of the entropy function is unchanged when taking the sup in 2.1. over the set of any reasonable law.

81

3. EXPONENTIAL BOUNDS FOR THE EMPIRICAL BROWNIAN BRIDGE We assume in this section that for some constants any

f

in F;

we set

U = v-u

and

u and

v, u < f < v for

F-u = {f-u, fEn

The fo 11 owi ng entropy conditions are considered : a) b)

d(2) (F-u) < U (2 ) e U (F-u) < 2 00

Using a single method we build upper bounds for (1.1.1) that are effective in the following two situations 1°) Observe that F.

nothing more is known about the variance over

In this case we prove some inequalities which are analogous to Hoeffding's ine-

qual ity [30]. 2°) We assume that This time our inequalities are analogous to Bernstein's inequality (see Bennett [5])

We randomize from a sample which size is equal to

rJ=mn

I n Po 11 a rd 's [44],

.;

Dudley's [20] or Vapnik and Cervonenkis [51] symetrization technics,

m=2

but here,

following an idea from Devroye [16] ,we choose a large m Effecting the change of central law : P->- PN with the help of a Paul Levy's 'V 'V type inequality, we may study P -P P -P where Pn stands for the instead of n N n randomized empirical measure. Choosing some sequence of - measurably selected

- nets relating to

PN whose mesh decreases to zero and controlling the errors committed by passing from a net to another via some one dimensional exponential bounds, we can evaluate, conditionally 'V

to

PN , the quantity

82

Randomization

Setting from [1.n]

N=nm (m into

[1,N]

is an integer), let w be some random one-to-one mapping whose distribution is uniform (the "sample w is drawn without

replacement"). The inequalities in the next two lemmas are fundamental for what follows 3.1- LEMMA. For any

in

I;

N

lR N , we set SN

L

UN = (max (1;.)) - (min (I;i)); 1
pos it i ve c ,lower bounds for- Log 1°)

2°)

n

Sn

L

CN

2

oN = and

I;i'

i =1

'V

(~

N L I;D i=1

i =1

(~

~(i )'

N L

i =1

2

I;i)

the followi ng three quantities are, for any 'V

Pr

(l~n -

2nc 2

7"N

nc 2

3°) These bounds only depend on

!;

through numerical parameters

(UN,oN)'

Bound 3°) is new; concerning 1°) (due to Hoeffding [30]) , Serfling's bound is better (see [46]) but brings no more efficiency when m is large. The proof of lemma 3.1. is given in the appendix. From now we write

for the randomized empirical process

1

n

L 8 . n i =1 xw( i )

The

inequality allowing us to study the randomized process rather than the initial one is the following:

83

'V

The random elements

lip n-PN II F and

110~IIF ~

Besides, whenever

2 P

E

p2 , the following holds: 'V

(1 - 2 ' 2 ) a En' for any positive

lip n-P II F are measurable.

> c)

Pr ( Ilpn-pIIF

and any

a

in

~ Pr ( IIPn-PNII F

]0,1 [ ,where

n'

n' > (1-a) N

c)

N-n .

=

For a proof of this lemma see [16] using Dudley's measurability arguments in [21] (sec. 12) .

Statement of the results. 3.3. THEOREM. ------The following quantities are, for any positive

1°)

a)

d(2) (F-u) U

if

+ 0n,F( 1) ( 1 b)

e(2) (F-u) U

if

= 1;

2

!...) 3 (d+n) u2

2°)

= s(~:2)

<2

if

d~2)

) exp (-2

increases from 0 to 2 so does

110~11

Suppose that a)

~

(when

~

0

(F-u)

=

F

o (1) n,F

2

2

2)

U

t k+r]

k

t

exp (-2

0n,F(1) exp (On,F(1) (0) where

and n , upper bounds for

,

2d

=

t

,

with 0

< U,

t2

u2 )

k).

then

2d , 2 (~f ~ (d) +n ( 1+ t- ) 3(d +n ) exp ( U 02 \

t2 + ..!:!..(3U+t)))

!n

b)

on,rr(1) where

exp(O

if

e0 2 )

(F-u)

=

s <2

,

r(1) (~)-~-n (!)2p-~+n + 5(!)2p+n) exp ~

n,r

_ 2~(4-d p - 4+d4-d

U

(when

0

~

0

\

2(i

increases from 0 to 2 so does 2p) •

84

The constants appearing in these bounds depend on (2 )

NU (.,F-u)

and of course on

F only through

n.

Comments.

d;2)(F) <

From section 2.2., the assumption

00

is typically fulfilled whenever

•

F is some V.C.-class with real density d. bound 1 a) is sharper than those of 1.3.1. , in another connection O(F,n) t 2nd in 1 a) is specified in the appendix.

Thus

0

)

the factor

0 )

In the classical case (i . e. bound 10

)

F is the collection of quadrants on

lR d ) ,

a) improves on 1.2.3. but is less sharp than 1.2.1. in the real case

moreover the optimality of 1°) a) is discussed in the aopendix where we prove that d-1 lim Pr (11\lnll F > t) ~ 2 ~ n->-oo i=O Suppose that F = Aa,d

then, from section 2.3. we have

respects, Bakhvalov proves in [4] that if on

[0,1]d

(2t2) i -i-!e;2)(F)

~ . In other

P stands for the uniform distribution

then: 1

Ilv n II F ~ C n2 -

a

a

surely

Thus we cannot get any inequality of the 1°) or 2°) type in the situation where eF)(F)

>2

•

The border line case ,

For any modulus of continuity ¢ , we can introduce a family of functions A¢,d in the same way as Aa,d by changing u ->- ua into ¢ and defining S as the greatest integer for which ¢(u) u-S->-O holds. u->-O It is an easy exercise, using Bakhvalov's method, to show that d

Ilv n II

> C (Log(n))Y

Aa,d -

provided that ¢(u) = u2 (log(u- 1 ))Y and P is uniformly distributed on [0,1]d 2 ) (A¢,d) = 2 and we cannot get bounds such as in theorem 3.3. Of course

ei

(*)

so, there is a gap for the degree of the polynomial factor in the bound 3.3.2°)a) between 2(d-1) and 6(d+n) •

85

But the above result is rather rough and we want to go further in the analysis of the families

Aep,1

around the border line.

Then the (2)-entropy plays the same role for

A

CI.,

concerning the Donsker

1

property as the metric entropy in a Hilbert space for the Hilbert ellipsoids concerhold~

ning the pregaussian property, that is to say that the following

:

1

is a functional P-Donsker class whenever

1

J (L09(N~2)

(E,Aep, 1)))2 dE <

00

•

o

is not a functional A-Donsker class whenever

(i)

ep(u) ~ (

1

(*)

u )2 !Log(u)[

follows from Pollard's central limit theorem in [44].

(ii) follows from a result of Kahane's in [32] about Rademacher trigonometric series.

ep ( u) = J ILo~ {u}I ' we have from [32] p. 66 that : En en(t) t + L belongs to Aep, 1 with some probability PK + 1, where n>1 KnLog{n} K->= is a-Rademacher sequence and en(t) = /2 cos (21Tnt) . In fact, if we set

Let us consider a standard Wiener process on

So that, with probability more than

(En)

L2 ([0,1]), we may write

PK ' the following holds 1

IIWII A ~ ~. L Iw(e n ) I nLog{n) ep, 1 n~ 1 By the three series theorem the series L IW(e n ) I nLog{n) almost surely and therefore W is almost surely unbounded on The same property holds for any brownian bridge some Wiener process provided that of

G.

So

Aep,1

W(1)

is some

G for

N(0,1)

f+G(f)

+

fW(1)

is

random variable independent

is not pregaussian and (ii) is proved.

(*) We write f ~ g ,when 0 < lim

diverges to infinity

1 (fg- 1 ) < lim (fg- ) <

00

86 /IAn upper bound in situation 2°) is also an oscillation conti'cl".

If we set

Go

=

{f-g

0 (f-g)

<0

n ,

, f,gE'

it is not difficult to see that:

N~~) (. ,Go+U) ~ ( NQ 2) (. ,F-u))2 U into

thus changing

situation 2°) hold with

2U

and

Go

d into

2d

instead of F,

if necessary the upper bounds in the constants being independent of

0

because of 3.4•. In particular if

•

F is a V.C.-class with real density d, we set A(o,n,t) = Pr ( Ilvnll Go

> t)

At it is summarized in [231 Dudley shows in [20] that r is small enough, 0 = 0 (ILO~(t)~ and n > O(t- ) with r

A(o,n,t) < t

whenever t

>8

Applying 3.3.2°) a) improves on this evaluation for then t A(o,n,t) ~ t whenever t is small enough, 0 = O( ILog(t) I ) and

n

~

0 ((

ILo~lt)(4)

In order to specify in what way the constant in

bound 2°)a) depends on

F,

we indicate the following variant of 3.3.2°)a) • 3.5. PROPOSITION. ----------NQ2) (E,F-U) ~ C (co d- 2d for any E in ]0,1[ and some Ilo~11 F ~ with 0 not exceeding U, then there exists some

If we assume that EO

in

]0,1 [that

i

E1 in ]0,1 [ depending only on that : Pr ( Ilvnll F > t)

~

EO

and

a constant

K depending only on

C such

2 2 K E- d (!:!U)-4d (1 + t 2 )14d exp ( t ) 1 o - 2(i + U(3U+t))

!n

From now on

L stands for the function

x-> Log(xve).

3.6. COROLLARY. --------Let (On)

• (F n ) be some sequence of V.C.-classes fulfillin9(M) with entire densities Then (with the above notations)

t ; whenever

o~

=

o( 1/(D nL(D rl )))

and

Pr ( Ilv II G n on 0~2 = 0 (In) •

> t)

-+

n+oo

0 for any positive

87

(Provided that

Dn

0({6) , such a choice of a

Ln

n does exist).

Comment.

According to Le Cam [38] (Lemma 2) and applying 3.6. the process admits finite dimensional approximations whenever

Dn =

(Lo;~n))

0

{vn(f), fE Fn} and provided

that Le Cam's assumption (Al) is fulfilled. This result improves on Le Cam's corollary of proposition 3 where

Dn = O(n- Y)

1 for some Y < 2 is needed.

Proof of 3.6.

• F be a V.C.-class with entire density

Let

D and real density

Using Dudley's proof in [20] (more deta i 1s are given in [40])

d .

it is easy to

show that, for any w > d (or w -> d if d is "achieved"), we have r~ (2) (E, F)

< K1+(1/2I Lo gEI)

1

for any

exp (2w) (1 + 21 LOgE I) w E-2w 3 D K = 2D! (2D)

E in ] 0, H , with in particular when w=D ,

So from Stirling's formula we get N;2) (E,F) tant

C . 1

~ C~ e 5D 23D E- 4D for any E in

Hence, for any

E in

] 0,

1.] 12

]0,1 [ we have

and some universal cons-

NF) (E,n ~ C~ (2e)5D E- 4D thus, applying 3.5. to the class We propose below

G yields On

another variant of

3.6. inequal ity 3.3.2°)a), providing an

alternative proof of a classical result about the estimation of densities. 3.7. PROPOSITION.

-----------

If we assume that some positive bound of

2 d0 )(F-U) = 2d <

00

and

Ilo~11 F ~

V , then there exists some positive constant

Pr ( Ilvnll F

°V,F,n(1~

> t)

(n(1 +

is, for any positive

:~))3(d+T1I;

i

with

3.3.2°)a) •

In - -

for

C such that an upper

t, given by :

n (O)_4(d+ ) exp (- 2

\i

+ ..t(CLLn (..!:!.. + o)+t)))

;;; In the situation where

uv
;;;

U is large this inequality may be more efficient than

88

Application to the estimation of densities : minimax risk.

Let

lR k ;

KM be the fo 11 owi ng kernel on

KM(y) = l/!(y'M y ) for any y where I/! and

in

lR

k

is some continuous function with bounded variation from

M is some

lR into

kxk matrix.

Pollard shows in [45] that the class 2

K~{KM(.-x), MElR k

,xElR k }

• is a V.C.-class of functions and so : N1(2) k,K)

~

CE -w

for any

E in

]0,1 [

where

C and w depend only on

k.

Now if we assume that P is absolutely continuous with respect to the Lebesgue measure on lR k , the classical kernel estimator of its density f is :

fn(X) = h- k Pn (K(·h x)) where

K is a KM with fixed

I/!

and

M so that

I

K2 (x)dx < 00

Proposition 3.7. gives a control of the random expression k xElR},o

So, if we assume that

for any E

tin

[1 + oo[

C and

(Jhk Dn) ~ T + O(n a ) T6 exp

and 6 .

[1,

+ 00[,

provided that

We choose T = O(/Cn) , thus:

Dn

nh k > 4 62 •

= sup lfn(x)- f(x)] : x

Hence, after an integration:

- --,;-2---'-T2......--~T~ 2(C + 0 (L_L_n_) + _ )

Jhk

for any T in

-f by choosing;

n

U = h- k/ 2 where C2 > llfll oo IK 2 (X)dX.

1l > h- k > C2 , we get, setting Ln -

and some positive a

f

•

Jhk

89

Provided that

f

belongs to some subset of regular functions 8 , the bias

f-f can be evaluated so that the minimax risk associated to the uniform distance on R k and to 8 can be controlled with the same speed of convergence as

expression

in [29] , via an appropriate choice of

h .

3.8. SKETCHES OF PROOFS OF 3.3., 3.5., 3.7. (More details are given in [42]) . First, by studying the class G u=o

and

Let us proof theorem 3.3 ..

v~1.

parameters such as and positive

a, and

S,I3

We set :

I.l

( = -t

Pr(N)r.)

All along the proof we need to introduce

(in ]0,1 [) ; r, m (in }j) ; a (in ]1,+oo[); q (in ]0,2])

In

We write

(' =

and

Pr (1lvnll > t)

'V

Pr ( II Pn - PNII > E')

(1 -

1)

m

(1-a)

(

.

for the probability distribution conditional on

and 11·11 instead of 11·11 F

A bound for

instead of F , we may assume that

which are all chosen in due time.

y

N = mn

(x 1 , .. • ,x N)

f-u fE F} {iJ'

~

for short.

will follow, via 3.2., from a bound for

which is at first performed conditionally on

(x 1 ,··· ,x N)

The chain argument.

Let

be a positive sequence decreasing to zero.

For each integer of 2.4).

j

a ,.-net J

A projection '1 j 2 2 holds.

PN( ( 11 j f- f) )

i

F.

J

can be measurably selected (with the help

F.

F onto

may be defined from

J

so that

'j

Then 'V

'V

II (p n - PN) So, if

0

(Id -

'1

r )11 i

L j~r+1

II (p n - PN)

is a positive series such that

(nj)

L

j>r+1

o

(,1. - 11· 1) II JJ

n·J < I.l

'V

Pr(N) ( IIPn-PNII where

A and A B

B are the

Nr IIPr L j~r+1

(N)

> (') < A + B

(x 1 , ... ,xN)-measurable va ri ab1es 'V

(I(Pn-P N) a 11rl ? (N) 'V Nj IIPr (I(Pn-P N)

> (hI) (')11 0

,

(,1r 1f j_1)I > nj c:' ) II

we get

90

A is the principal part of the above bound and

B is the sum of the error

terms. Inequalities 1°) or 2°) of Lemma 3.1. are needed to control

A according to

whether case 1°) or 2°) is investigated. B , giving

Bound 3°) in Lemma 3.1. is used to control B

<2

L j~r+1

nj = (j_1)-a and

N~ exp (J

\

,2 2 n En. \ J

4m

:5.8. 1.

T~ 1)

r

L n· < ~ holds whenever j>r+1 J the control of the tail of series 3.8.1. is performed via the following

Choosing a > 2),

r

=2

(so

+

elementary lemma 3.8.2. Lemma.

Let lji : [r, +00[

->-

lR.

Provided that

~I

is an increasing convex function, the

following inequality holds: exp (- lji(j)) ~ ~ exp (-lji(r)) d stands for the right-derivative of lji L

j~r+ 1

where

ljid

We choose S

= 1 under assumption a) and S

(~)

under assumption b) •

Proof of theorem 3.3. in case 1°). We choose

a

=

t- 2 ,

m = [t 2 J and

T. J

= Jl j-(a+S) and apply 3.1.1°), then In

A < 2 Nr e 10 exp (_2t 2 (1-2~)) Under assumption a).

Considering the type of inequality we are dealing N. ~ C t 2d }(a+1ld (instead of N. < C' t 2d ' }(a+1ld' J

J -

We c hoose,

"~ -- t- 2

an d a

=

with we may assume that for any

Max (2 , 1 + 3n 4d) ' so

A ~ 0n,F(1) (1+t 2 )3(d+ n ) exp (-2 t 2 ) and

d' > d) •

pIIIN_a.s.

91

P!liN -a.s. whenever

t 2 > 7+4d(a+l)

Now the above estimates are deterministic, so using

Lemma 3.2., theorem 3.3. is proved in situation 1°)a) • With the idea of proving proposition 3.5. note that, setting a method gives, under the hypothesi sin 3.5. , that

with

~

2 , the above

Pr( llvnll> t) is bounaed by:

K (E~1 t)4d (2+t 2 )12d exp (_ 2t 2) 1 K1 depending only on C, whenever t 2 > 7+12d.

Under assumption b).

We may suppose that We set

I.l =

N.

< exp

(C t1; jc;(a+S))

J -

t -2y , where 1;(1+2y(~))

~

n and for y ~ y( 1;) - 2 S > 1 to ho 1d, whe re

tion when a

~

+00

(Namely:

2( 1-y( 1;))

~

2( 1-y) , then we choose y( 1;)

k).

a 1arge enough

is the solution of the above equa-

So:

2 k A ~ 0n,F(1) exp (On,F(1) t +n ) exp (-2 t ) and B

whenever

n, F( 1)

0

eXfJ

(-2 t 2 )

t 2 ~ ~ + 5 + C t1; 22S+2 •

So theorem 3.3. is proved in case 1°). Proof of tneorem 3.3. in case 2°). We set

Q

~!

o

The variable

and choo'se m =<.:1 q, a ~ 2\fq ,

I.l

~ <.p-q and

,. J

2

0p •

In fact, let 8N be the

(x 1, •.• ,x N)-measurable event:

8N ~ {llo~ - o~11

where

> s}

o~(f) ~ PN(f2) - (P N(f))2 for any f in F .

Each terw of the following estimate is studied in the sequel

A'

E(AlI

In

A is this time controlled with the help of 3.1.2°), so now the

probZem is to repZace

where

= .Q.

c)

GN

and

BI

~

E (B 11 8 c) N

92 Bounding

is a r1'Oo7.e", of tUDe 1 U)

Pr (eN)

F2 = {f2, f E F}

' For, se tt lng

2

, we have

2

liON - opll~IIPN-PIIF2 + 21 PN-PII Since

N;2)

(.,F 2 ) ~

in 3.3.1°), so, choosing

N;2) s =

1

(~,F)

2 F

and

fulfills (M), we may use the bounds

2

we get: Pr (eN) ~ Co exp (- ~)

k, 12nm

The evaluation of A I and B I .

2

Ii0NII

A'

~

2

~ a +s

c

holds on

eN

thus applying 3.1.2°)

2 N exp (5 <.p2-q) exp (\ r

2 (a

2 ...:t:..._:--...,..,,-), t+ljJ1-q/2

2

+ (

gives whenever

))

In

Moreover

<2

B'

2 nE,2 LN. exp (- - 2 j~r+1 J 4 a

Now the proofs are completed as CJ. '"

Max (2,1 + 4d) 3n

large enough for

A'

and

1 - ~

2 -< p + nand S >

<2

E~2d

C

we choose

e S 0-2d

2s

)

in case 1°), choosing this time

under assumption a)

To prove proposition 3.5

(j-1l

q = (2-1;)(1 +

,,(~~r)

q=2

and

+ 1)-1 \\ritn

CJ.

to hold. CJ.

= 2, so

<.p2d (2+(jJ2)6d exp ( _ _----"t,-2--"1.".--,-,,\

\

2(i+(3+t)J'

In B whenever

<.p2

~

'

< 2 C2

-4d -4d 4d 2 12d ((<.p2_ 8 )2\ EO a <.p (2+<.p) exp \4 )

8 + 12d

Besides, using 3.8.3., we get: Pr (eN)

~

E

2 K 1

(~)-4d (~)4d

(2 +

2 2 12d ) exp (- ~)

~

2

whenever

~ ~ 7+12d,

which completes the proof of proposition 3.5

via Lemma 3.2.

Proof of proposition 3.7. We assume that

u=O

and

v=1.

Inequality 3.3.2°)a) may

be written

93

2

whenever

:L2 -> 5

IJ Defining the following sequences by induction 2 b. 1 44d+1 a. (2 5 +~ + 2 ) a·J+ 1 J nIJ IJ 2 (.1 + lb.) bj +1 J Iii Iii 3 with a ,,1 and b we call M. the following inequality: 0 o Iii J 2 -a t 2 a2 t2 (IJ 1 !_) :L > t) < Ka. t ) whenever exp Pr ( Ilv n II \ 2 2 -> 5 J \ 2(IJ 2+b. + -) IJ IJ J In The, assuming that same way as 3.3.2 C )a) Then inequalities J

1 + [~]

M. holds, it is possible to deduce M. 1 from M. by the J+ J J from 3.3.1°)a) (technical details are given in [42]) (M j ) hold by induction.

Using inequality MJ ' where and a few calculations yield proposition 3.7 . .

4. We assume that section still hold

P(F 2 ) <

EXPONENTIAL BOUNDS FOR THE BROWNIAN BRIDGE

00.

We want to show that the bounds in the preceding

for the brownian bridge.

4.1. THEOREM

If e~2)(F) < 2 , then there exists some version

Gp of a brownian bridge rela-

ting to

P whose trajectories are uniformly continuous and bounded on (F,pp) . (2) . 2 2 2 Moreover, setting l; ~ e (F) , 1f IIIJpl1 F i IJ i P(F ), an upper bound for F Pr ( IIGpII F > t) , is, for any positive t and 11 , Siven by

o

11,

F(1) exp (0

11,

or, if more precisely

F) IJ-l;-211 ( P(F 2 ))l;/2+ 11 (!)2p-l;+I1+ (!)2p+l1) exp (- t~) IJ IJ 2IJ 2d <

00,

by

4.1.1.

94

4.1.2.

where

p is defined in the statement of theorem 3.3.

Comments.

In the framework of theorem 4.1. the existence of a

regul ar

version of a

brownian bridge is an easy consequence of the proof of 4.1.1., but is of course a \-lell known result (see [18]).

Moreover the bounds in 4.1. 'ire in thi s case sharper

than the more general Fernique-Landau-Shepp inequality (see [25])

that

can be

written Pr (

IIGpII F > t) i

2

t C(a) exp (- - )

20 2

Proof of theorem 4.1. If

is countable

l'

The calculations are similar to those of the proof of theorem 3.3. but here of course a sequence of nets in

(1',1') relating to P is directly given.

Moreover

the following single inequality is used instead of Lemma 3.1. : 4.2. LEMMA:

Let

V be a real and centered gaussian random variable with variance

v2

then Pr

(IVI > s) i

s2

2 exp (- 2i) for any positive s.

The choice of parameters being the same as in the proof of 3.3.2°) (except 1·2

J

2 0 --2-

J.-2(a+s)) ,

P(1' )

d 4.1. 1. an d 4 .1 .2. are prove.

S'1nce 4 .1.1. 1S . a 1 so an

oscillation control, the almost sure regularity of Gp follows from Borel-Cantelli. The general case.

Since

(F,Pn) r

is separable

, the familiar extension principle may be used to

construct a regular version of brownian bridge on on a countable dense subset of F. this version.

l' from a regular version defined

Inequal ities 4.1.1. and 4.1.2. still hold for

95

Comment.

The optimality of bound 4.1.2. is discussed in the appendix. the polynomial factors are different in 3.3.1.2°)a)

The degrees of

and in 4.1.2. ; the reason is

that bound 3.1.3°) is less efficient than bound 4.2.

5. WEAK INVARIANCE PRINCIPLES WITH SPEEDS OF CONVERGENCE P(F 2+o) <

We assume from now that

00

for some

0 in

]0,1].

Using the results in sections 3 and 4, we can evaluate the oscillations of the empirical brownian bridge and of a regular version of the brownian bridge over F , so we can control the approximations of these processes by some

Ek-valued processes

(where Ek is a vector space with finite dimension k). The Prokhorov distance between the distributions of these two processes is estimated via an inequality from Dehling [15] allowing reasonable variations of

k with

n

Oscillations of the empirical brownian bridge over F .

n over F are controlled with the help of a truncation (the proof in this case is straightforl'.'ard ) on the one hand and of

The oscillations of from 3.3.2°)a)

V

a slight modification in the proof of 3.3.2°)b) (truncating twice) on the other hand. We

shall not give any proof of the following theorem (the reader

will find it

in [42]) .

~o

We set

=

P(F 2+o)

If we assume that

then an upper bound fJr Pr (1IvnIIF> t) given by a)

d~ 2)

If

7d

OF (1) n

(F) = 2d

t Sd

(0.1

(

exp \.-

<

00

,

si

whenever the following condition holds:

2 2 2 Ilapll F ~ a ~ P(F)

is, for any positive

t

with

such that

/ria t

~ 1,

/i~1,

2

96

5.1.1.

b)

+

(p

e (2) (F)

If

0(1)

F

lJ

-o/2

on

=

s <2

~-2ot-2+o

for any positive n

v

is defined in the statement of 3.3.) whenever 5.1.1. and the following hold n0/ 4i+ o

> 512

lJ

5.1.2.

o

Remark.

Note that Yukich in [54] also used KOlcinskii-Pollard entropy conditions to prove analogous results to theorems 3.3. and 5.1., but our estimates are sharper because of the use of randomization from a large sample

1S

described in section 3

Speed of convergence in the central limit theorem in finite dimension.

We recall below a result that is due to Dehling [151 (the first result in the same di rection is due to Yurinski i [53]). 5.3. THEOREM. ------Let

(Xi )1
be a sample of centered

lR k -valued random variables.

We write

Fn for the distribution of the normalized sum of these variables and G for the centered gaussian distribution whose covariance is that of X . 1 k Let 11.11 2 be an eucl idian pseudo-norm on lR and '12 be the Prokhorov distance that is associated to

11.11 2

If

E ( IIX111 ~+o)

= IJ

<00,

then

o

- 8 14 '12 (Fn,G) ~ K n k / 1J1/4 (1 + lL(n- o!2 k- 1 1J)1 1/ 2 ) where

K is a universal constant.

Weak invariance principles for the empirical brownian bridge.

In order to build some regular versions of brownian bridges with given projection

97

on a finite dimensional vector space (or further in section 6 on a countable product of such spaces), we need two lemmas. ~~~~~

5.3.

(Berkes, Philipp [6]) .

R1 ,R 2 ,R 3 be Polish spaces, and be some distributions respecti°2 °1 vely defined on R1xR 2 and R2xR 3 with common marginal on R2 . Then there exists a distributions on R1xR 2xR 3 whose marginals on R1xR 2 and R2xR 3 are respec° tively 01 and 02 Let

l~(F)

Remember that

is generally not separable.

The following lemma is fun-

damental to avoid this difficulty (see [23]) . The space 0 to be mentionned below is defined in Section 2. ~~~~~

5.4.

(Skorohod [48])

Let ginal is

R1,R 2 be Polish spaces and be some distribution on R1xR 2 with mar° q on R2 If V is a random variable from 0 to R2 whose distribution

q , then there exists a random variable

tribution of

(y , V)

is

y

from

0

to

R1 such that the dis-

°

Concerning our problems of construction the point in the sequel is that the distribution on

l~(F)

of a regular version of a brownian bridge is concentrated on

a separable space. Now we can state some weak invariance principles for the empirical brownian bridge with speeds of convergence. 5.5. NOTATIONS. --------From now y and [0,1] x lR+

and

B are positive functions that are respectively defined on

[0,2] y(x,y)

by: =

8 + 2~ (4+x)

and

where, as in the statement of theorem 3.3.

2(1-e(z)) B(z) = z(2-2p(z)+z) p(z)

2z(4-z) 4 + z(4-z)

98

5.6. THEOREM Under each of the following assumptions there exists some continuous version on such that Pr ( Ilv n _G(n) II F -> a n)<Sn P are defined hereunder (we reca 11 that FE L2+6 ( P) and that

of a brownian bridge relating to where

(an)

and

(Sn)

P, G(n) p

e~2) and d~2) are defined in Section 2) : a)

If

for any

d(2)(F)

=

F

2d <

< y( 6, d)

L

If NF(2) (E, F)

a' )

00

~ C

-1 d E-2d (LE) for any

E

in ]0 , 1 [

I; < 2

b)

for any

< S(I;)

L

and any positive

s.

Proof of theorem 5.6. Let of

a be an oscillation rate

F on a a-net

F(a)

We approximate Setting d into ted.

2d

Ga~

(de~endin~

relating to

~

F by vnoITa

a} , we may apply theorem 5.1. to Go

if necessary), hence the quantity

IIVn-VnoITaIIF~llvnIIG

Writing

a

1100

random variables

wh ere

B

for the Prokhorov distance associated to 11·11 F(a)

vn(a)

such that

~ 1l ( F 00

can be evalua-

Fn,a be the distribution of vn IF( a ) on the k-dimensional loo(F(a)) and G be the corresponding gaussian distribution.

Strassen's theorem [49] , there exists a probability space

loo(F(a))

(changing

a

Besides, let

vector space

on

be a projection

P

vn uniformly over {f-g , rp(f,g)

on n) and 10

n,a , Ga ) .

and

G(a)

(~'

and applying

,A' ,Pr')

with respective distributions

and two

Fn,a and

Ga

99

So, using lemma 5.3. , we may ensure the existence of some continuous version of a brownian bridge

Gp relating to

applying lemma 5.4. with rl

V: w->-vn!F(o) , we may assume that

pr( !lvn!F(ofGp!F(o) II F(o) ~ B)

with

G(o) = Gp!F(o)

p such that

Hence, noticing that

i

and then,

Gp is constructed on

B.

Ilv n IF(ofGpl F(o) I! F(o)

=

!! (v n-G p)oJ1 o !! F'

we get

Pr ( I!vn-Gpll F > 2t+B) i A+B+C where

A" Pr ( Ilvnll G a

> t)

and

C" Pr ( !IGpll G a

Theorem 5.2. is used to control k ~ N~2) (~,F)

noticing that

> t)

B (with II·!! F(o)

.

i

I!·!! 2 i Ik I!·!! F(o))

according to remark 2.6 . .

Moreover C is evaluated with the help of theorem 4.1., so the calculations are

co~pleted

via an appropriate choice of

t

and

o.

6. STRONG INVARIANCE PRINCIPLES WITH SPEEDS OF CONVERGENCE.

The method to deduce strong approximations from the preceding weak invariance principles is the one used in [43] to prove theorem 2 : the weak estimates are used locally, giving strong approximations with the help of maximal inequalities and via Borel-Cantelli lemma. Maximal inequalities.

As was noticed in [23], the proofs of the following inequalities may be deduced from the one given in [10] and in [32] . Notation.

We set

X.

J

6x. - P for any integer J

j

100

6.1.

~£~~~ (Ottaviani's inequality).

k L X. .

We set Sk" j

c ~ max k
where

=1

Then, for any positive a, the following inequality holds:

J

(1-c) Pr (max IIS k II F k a) .

> ra) < Pr ( IIS n II

F

> a)

More precisely, for symmetrical variables, the following sharper inequality is available 6.2.

~£~~~ (Paul Levy's inequality).

Let

(Yi)1
bles where

(B, 11.11)

is a normed vector space.

Pr (max Ilskll > a) k
trical then where

be independent and identically distributed B-valued random varia-

i

If we assume that

Y1 is symmeholds for any positive a ,

2 Pr (11Snll > a)

S = L y. k j=1 J

Strong approximations for the empirical brownian bridge. 6.3. THEOREM. -------

Under each of the following assumptions some sequence versions of brownian bridges relating to defined on a)

if

~

(Yj)j~1

P that are continuous on

of independent (F,pp)'

may be

such that : d2 )(F) = 2d < F

1

Iii

00

,

n

II L

j ~1

(x·-~·)II F J

a. s.

J

for any a < 2( 1Y(O,d) +y ( o-;crn (2) a' ) if, more precisely NF (c,F) < Cc- 2d (1+Ls- 1 )d for any c in ] 0 , 1[ , n 1 (X .-y.) II = O(n-y(o,d)/(2( 1+y(o,d))) ((Ln)(1/2)+d + (Ln)(5/4)+(d/2)))a.s. In II j~ 1 J J F

b)

if

e?) (F) = I; < 2 , 1 II n (X.-Y.)II

,Iii

j~1

J

J

F

"O(Ln-(6/ 2 ))

for any 6 < 6(1;) . Where

y(.,.)

and

s(.)

aredefinedin5.5.

a. s.

101

For a proof of 6.3., see [42]. Comments.

When passing from weak invariance principles to strong ones, the speeds of convergence are transformed as follows within our framework:

incase a).

in case b) Transformation (ii) appears in theorem 6.1. (under 6.3 .) from [23], it is not the case for transformation (i) in the same theot'em (under 6.4.). On the contrary transformation (i) is present in finite dimensional principles and appears to be optimal in that case: more precisely, the rate of weak convergence towards the gaussian distribution for 3-integrable variables is ran' 1S rang1ng . a bou t n- 1/ 6 (see gl'ng about n- 1/ 2 when th e ra t e 0 f s rong t convergence [39] for the upper bound and [9] for the lower bound) lin the real case Application to V.C.-classes.

Applying theorem 6.3. with 6 =1

in the case where

v

F is a V.C. -class wi th

real dens ity d, we get a speed of convergence towards the brownian bridge that is 1 O(n-O:) for any This improves on 1.3.3. but is less sharp that 1.2.4. 0: < 18+20d in the classical case of quadrants in lR d

Following an idea from Dudley in [21] (sec. 11), the study of the general empirical processes theoretically allows one to deduce some results about random walks in general Banach spaces. metric space (5,K)

As an application of this principle

and the space

ped with the uniform norm

11.11

C(5) •

00

equipped with the Lipschitz-norm:

Let

let us consider a compact

of real continuous functions on

5,

equip-

X be the space of Lipschitz-functions on

5

102

II· II L : x

->-

II x II + tfS sup 00

We write

N(E,S,K)

ds,t) > E for any

sft

Ix(t)-x(s) I (s t) K,

for the maximal cardinality of a subset in

R of S such that

R.

We may apply our results through the following choices: F=II.II L ·

F={oS,sES}and Then

(X, 1/.11

) 00

is a Suslin space (but is not Polish in general), so

fills (M) . Moreover, for any distribution Q( ( Os - 0t )2)

so

N~2)(.,F) ~ N(.,S,d . Besides

11.11

Therefore, considering a sequence tributed

00

= II.II

2 Q(F ) ,

.

F

(Xj)j~1

of independent and identically dis-

C(S)-valued random variables such that: IX1(s)-X1(t)I~Mds,t)

with

p~2)(X) we have:

Q in

~ K2(s , t)

F ful-

E(M 2+0) <

00

and

E(X~+O(to)) <

00

for one

,

for any s,t

to

in

in

S.

S, we can apply some 5.5. or

6.3. theorem to get speeds of convergence towards the gaussian distribution, whose structure depends on

N(.,S,K)

(the central limit theorem for such uniformly

Lipschitzian processes as above is due to Jain and Marcus in [31]) .

A P PEN D I X

First let us recall Hoeffding's lemma

(see (29]) .

Hoe[[ding's lemma.

Let

S be a centered and

[u,v] -valued random variable, then 2 2 E(exp(tSj) ~ exp (t (~-u)) , for any t in R •

We may assume that w is chosen as follows: drawing - with uniform distribution - a partition] = (J i )1
such that

IJil =m

103

tion -.

J. - with uniform distribu-

independently in each

. then, drawing an index w(i)

1

The following evaluations are conditional on ]

but the last bound will

not depend on J, 'V giving 3.1. 5

We set

Z

5

iln - IfN and we write

~

A for the logarithm of the conditional La-

place transform of Z Then setting

-

si

~

we have , for any

m

in

s

lR

A(s) then, since the logarithm is a concave function A(s) ~ n Log (~ where, writing

ON

s

N L

j =1

5N

exp (ri (Sj - W))) ~ n AN(~)

for the uniform distribution on

the logarithm of the Laplace transform under

ON

{S1""'SN}

of x

AN

~

x-EO (x).

5'

N

stands for

Therefore the

5'

Cramer-Chernoff transform of Z is larger than that of : - EO~n (:)

under

O~n

N

where

5'n stands for the sum of n i.i.d. random variables with common distribution Then, Hoeffding [29] and Bernstein

[5]

inequal ities yield 3.1.1°) and 3.1.2°)

In order to prove 3.1.3°) we may assume that 5N = 0 (otherwise changing 5

Sj

into

N

Sj - W) Then, applying Hoeffding's lemma to the conditionally centered random variables and setting

u. = min 1

l.(t) 1

jEJ

S. i

and

v. = max S. , we get: 1

J

jEJ. J 1

(v.-u.) LOgEJ(exp(t(s(.)-~.)))< 1 1 Xl 1 8

2

t

2

,foranytinlR.

Hence A(s)

2

1 . (~) < _s_ 1 n 2 8n

n

L

i =1

(v.-u.) 1

2

1

-

and therefore s2

2

A(s)~lfrlm0N

yielding 3.1.3°) via Markov's inequality.

s2 <2 4n

n L

L

i ~ 1 j EJ.

1

104

We use Goodman's work in [28] to give a lower bound of the probability for the supremum of a brownian bridge to cross a barrier. Notations.

of

We set I = [0,1] and write for any integer d, 1 for the element d d d lR Moreover, for any s in I , we set p(s) = sl' .. sd .

(1, ... ,1)

A.1. Theorem. Let

d be an integer and

Wd be some standard d-dimensional parameter Wiener

process, then, on the one hand Pr (sup Wd(s) < t IW d (l d ) sEI d -

(i)

for almost any real number hd(a, t)

=

1

+

a

=

at) ~ hd(a,t)

(in Lebesgue sense) and any positive

exp(2t 2(a-1))

t, where

d-1 . ( 2( ))i L (_1)1+1 2t . ~-1 I ( ) i=O 1. J-oo,1] a

and on the other hand : (i i )

Proof of theorem A.1. If d=2

the whole proof is contained in [28] .

Otherwise, proceeding exactly as in [28] yields the following inequality: Hd(a,t) = Pr(sup Wd(s) sEId where

o

~ t IWd (l d )=at) ~ J (1-exp(2t 2r)) dF t ,d_1 (a,dr) A.2. a-1

Ft ,d_1(a,r) = Pr(W d_1(s)-rtp(s) ~ t, VSEI d- 1 We want to proceed by induction.

It is enough to notice that:

A.3. Lemma.

for any integer

k, any positive a and almost every B in

lR (in Lebesgue sense).

105

Proof of A.3. W is a regular gaussian process, it is enough to show that the expeck tation and covariance functions of the processes Wk(.)+p(.) and Wk(.) are the Since

same conditionally to respectively W (l k )+a and k

Wk (l k) .

Since Wk is gaussian, E(Wk(s) IWk(lk'=y) and E(Wk(S)Wk(s' )IW k(l k )=y) respectively linear and quadratic functions of y, then the knowledge of

are

E(W~(lk) W~(s) W~(SI)) with l+m+n ~ 4, yields E(W k(S)IW k (l k )=y) = p(s)y , E(Wk(s)Wk(s')!Wk(\)=y) where

p(s)p(s')/ + (s"s') - p(s)p(s')

SAS'

A.3.

Let us return to the proof of A.l .(il Using lemma A.3., we get:

so

Then, integrating by parts, inequality A.2. becomes: 0 Hd(a,t) ~ Ft ,d_l(a,O) - lexp(2t 2r)F t d_l(a,r) la-I + 2t 2 fO exp(2t 2r)F t ,d_l(a,r) dr , a-I

hence Hd(a,t) ~ 2t 2

o

exp(2t 2r) hd_1(a-r,t) dr a-l But an easy calculation yields: Jr

O

2t 2 Jr exp(2t 2r) hd_1 (a-r,t) dr = hd(a,t) a-I Therefore A.l. (i) is proved by induction (it is shown in [33] p. 284 inequa 1ity A.1. (i) holds when d=l) . In order to proof (ii), we notice, following [7], p.84, that

Pr(W d E-/ 0 ~ Wd (l d ) converges weakly in

~£

)

C(I d ) towards the distribution of the brownian bridge

that

106

Wd - pt. )W d (1 d ) whenever

E converges to

°.

So, inequality (i) gives Pr (sup d SEI

Wd(s) - p(s)W d (1 d ) > t)

~

1 - hd(O,t)

therefore (ii) is proved. Comment. 'V

Theorem A.1. was proved by ourself (see [40] and [41]) but also by E. Cabana in [11] * In another connection, inequality A.1. (ii) ensures that some polynomial factor t 2h (d)

w,"th

h(d»d-1 _

b ds 331°)) canno tb e remove d", noun .. a an d412 ..•.

The calculations yielding 3.3.1°)a) are slightly modified here, where the entropy condition a) is replaced with a more explicit one. A.4. Theorem. If we assume that a'l

F is

[0,1]-valuedandthat -2 NF)(E,F) ~ K1+1/ Lo g ( E 1 (1+LOQ(E- 2 ))d E- 2d

then, an upper bound for

Pr (

Ilv n II

F > t) is, for any

for any t

in

E in

]O,n

[1,+00[, given by

4H(t) exp(13) exp(-2t 2 ) + 4H 2 (t) exp(-(t 2-5)(Lt)2) where

Proof of A.4. Lt 2 + 1 , then In the proof of 3.3. 1°)a) we choose a = ----2 LLt A ~ 2H(t) exp(13) exp(-2t 2 ) P!liN -a.s. B ~ 2H 2(t) exp(-(t 2-5)(Lt)2)

whenever

*

t

2

Thanks to

~

P!liN -a.s.

6+4d , yielding A.4. via lemma 3.2.

M. Wchebor and J. Leon for communicating this reference to us.

107

Comment.

Assumption a'l is typically fulfilled whenever case of

d

F

•

is a V.C.-class.

In that

may be the real density of F (if it is "achieved") or the integer density

F (see the proof of 3.6.).

REFERENCES

V-i.Me.~on,

[7]

ALEXANVER, K. (7982). Ph. V.

[2]

ALEXANVER, K. P~obab~y ~ne.qua£LtZ~ nO~ e.m~c.al p~oc.~~~ and a taw On Ue.M.te.d togMdhm. Anna£~ 06 Ptwbabildy (7984), vol. 72 ,N° 4,7047 -7 06 7.

[3]

ASSOUAV. P. Ve.lUUe. e.t

[4]

BAKHVALOV, N.S. On appMuma;te. c.a.£c.ula.t<.on 06 multipte. ~n.te.g~aU (~n RUM~n). V~tn~Q Mo~Q. Se.~. Maj". Me.R.h. MUOH. Uz. K~m. (7959) i, 3. 78.

[5 ]

BENNETT,

233-282.

G.

bt~.

~me.lU~on,Aml.

MaM. IlUt. hc.h. (7982).

lust.

FoU!U~, G~e.n(lbte. 33, 3 (7983)

P~obab~<-.ty ~ne.qua£LtZ~ 60~ ~~ Am~. Sta.t~t. A~Mc.. (7962),

J.

06

'ii,

~nde.pe.nde.n.t ,~ndom v~a­

33-45.

[6]

BERKES, I., PHILIPP, W. App~ouma.t<-on the.o~e.~ 60~ ~nde.pe.nde.n.t and we.aRty de.pe.nde.n.t ~andom ve.C.to~6. Ann. P~obab~<-.ty (7979), i, 29-54.

[7]

BILLINGSLEY, P. Conve.~ge.nc.e.

[8]

BORISOV, I .S.

Ab~.tMc..t6

06

~n6e.~e.nc.e., Budap~t

06

pMbab~y me.MM~.

the. CoUoq~um on non (7980), 77-87.

Wile.y, Nw Yo~Q.

paJ1.OJrl~C. ~.ta.t<-stic.al

[9 ]

BREI MAN, L. On the. .:i:t::LU be.hav<'M On ~u~ 06 ~nde.pe.nde.n.t'Landom z. WaMc.hun. VVlW. Ge.b. (7967), ~, 20-25.

[ 70]

BREI MAN , L. P~o bab~y. Re.a~ng MM ~.

[ 77 ]

'" E. On the. CABANA,

06

.tMlU~tion de.lUay W~e.ne.~ p~oc.~~ wUh one. b~~.

Ad~ 0 n-W ~ te.y

a

v~abt~.

(7 968) •

multi~me.lU~onal pa.Mme..t~ P~ob. (7984), ~, 797-200.

J. Appt.

[ 72]

'" E., WSCHEBOR, M. The. CABANA, S~mon BoL<'VM.

[ 73]

COHN, V.L. (7980).

[74]

CSORGO, M. , REVESZ, P. A nw me.tllOd to p~ove. S~Hn type. ta.w~ 06 ~nv~anc.e. p~n~pte. II. Z. WaMc.hun. Ve.~. Ge.b. (7975), ~,

.two-p~me.te.~ B~ow~a.n b~dge..

Me.MU~e. the.o~y. BiAR.haUse.~, Bo~ton

Pub.

U~v.

(7980).

267-269.

[ 15]

VEHLING, H. umU the.o~e.~ nM ~~ on we.aRty de.pende.n.t Banac.h ~pac.e value.d ~andom v~abt~. Z. WaMc.hun. Ve.~. Ge.b. (7983), 397-432.

[76]

VEVROYE, L. BOUHd~ 6M the. u~tlotun dev {at~olU 06 J. of MUU.,'VM. ArlcU'.. (7982), ]1, 7~· 79.

[17]

INCHI HU, A unA 6Mm boand folt the. .tail p~ubab'{LUlf 06 Ko.fmogMov-SmiAnov ~ta.t{6tiC.~. The. Anna.e~ On Stctt~~.t.{,C.6 (7985), Vot.73,N°2, 827-826.

('Jl1:)~It.<.cc<£ me.a~M~.

108

[78]

DUDLEY, R. M. The. -6J..zrv., 06 c.ompac;t ,6[Lbuu 06 Hlibe.Jtt -6pac.e. and c.on:Unu1;ty 06 GauMJ..an pIWC.rv.,-6rv.,. J. Func.tiona£ AnalY-6J..-6 (7967), ~, 290-330.

[79]

DUDLEY, R.M. Me:tJUc. e.nvwpy 06 Mme. &aMrv., 06 uu wah dJ..66vce.n:Ual bounc1aJL,trv.,. J. AppM xirrii:Ltion The.OI1.y (7974), !.!!-' 227- 236.

[20]

DUDLEY, R.M. Ce.ntlW£ bliUy (7978),

umU: ~,

the.OI1.e.mil 60f1. e.mpJ..Jr.J..c.al me.MUf1.rv.,. Ann. PMbai (7979), 909-977.

p. 899-929 ; c.of1.f1.e.c.:Uon

MCLthl?Jllatic.~

[27]

DUDLEY, R.M. SaJ..nt FtoUf1. 7982 • Le.c.tuJ1.e. /l:ote.-6 J..n

[22]

• DUDLEY, R.M., DURST • EmpJ..Jr.J..c.al Pf1.oc.rv.,-6rv." VapnJ..R-Ce.f1.Vone.nw c1.M-6rv., and PoJ..-6Mn pf1.oc.rv.,-6rv.,. PMba. and Math. Stat. (WIWc1.CLw) 7, 709- 775 (7987).

n07097 •

-

[ 23]

DUD LEY, R. M., PH I LI PP, W. InvaJr.J..anc.e. p!1.J..IWJ..ptrv., 6011. -6 umil 0 6 Banac.h -6 pac.rv., value.d f1.andom e.tvne.nt-6 and e.mpJ..Jr.J..c.al pf1.oc.rv.,-6 rv.,. Z. WCLM c.hun. Ve.f1.W. Ge.b • (7 983 ), E, 509- 552 •

[24]

DVORETZKY, A., KIEFER, J.C., WOLFOWITZ, J. A-6ymptotic. mJ..nJ..max c.hCLlWc;te.f1. 06 the. Mmpte. fu.tJr.J..buUon 6unc.:Uon and 06 the. &M-6J..C.al muttinomJ..a£. rv.,.tJ..mCLtof1.. Ann. Math. Stat. (7956), ii, 642-669.

[25]

FERNIQUE,

.!.i,

x.

R(gut~(

de.

pf1.oc.rv.,-6~

ga~-6J..e.I1-6.

Inve.nt. MCLth. (7977),

304-320.

[26]

GAENSSLER, P., STUTE, W. EmpJ..Jr.J..c.al pf1.oc.rv.,-6rv., : a -6Uf1.Ve.y 06 f1.rv.,utv., 6011. J..nde.pe.nde.nt and J..de.n:Uc.illy fuWbute.d :random vaJr.J..abtrv.,. Ann. Pf1.oba. i, 793-243.

[27]

GINE, M.E., ZINN, J. On the. c.e.n.tf1.a£. UmU: the.of1.e.m 60f1. e.mpJ..Jr.J..c.al pf1.oc.rv.,-6rv.,. Annal-6 06 PMbabliUy (7984), vot. 72, nO 4, 929-989.

[ 28]

GOODMAN, V. DJ..-6.tJr.J..buUon rv.,.tJ..matJ..OI1-6 6011. 6unc.:Uonal-6 06 the. .two paJUmie.te.f1. WJ..e.nvc pf1.oc.e.~-6. Annal-6 06 PMbabliUy (7976), vot. 4, nO 6, 977-982.

[29]

HOEFFDING, W. PMbabliJ...ty J..ne.quaUtirv., 6011. Wmil 06 bounde.d Jtandom vaJr.J..abtrv.,. J. Amvc. S~t. A-6MC.. (7963), 58, p. 73-30.

[30]

IBRAGIMOV, I.A., KHASMINSKII, R.Z. On the. non-paf1.ame:tJUc. de.fM-<-ty rv.,:Umatrv.,. Zap. Nauc.ha. Se.mJ..n. LOMI 708, p. 73-87 (7987). In R~-6J..an.

[37]

JAIN, N., MARCUS, M.B. Ce.ntJr.at umU: the.of1.e.m 60f1. C(S)-value.d f1.andom vaJr.J..abtrv.,. J. Func.:Uona£ Ana£Y-6J..-6 (7975), .!i, p. 276-237.

[32]

KAHANE, J.P. Some. f1.andom -6e.!1.J..rv., 06 6unc.tiol1-6. Le.xington, MM-6. D.C. He.uth ( 7968) •

[33]

KARLIN, S. TAYLOR, H. M. A 6-<.Mt C.OUMe. J..n StOC.hMtiC. Pf1.oc.rv.,-6rv., (7971). Ac.ade.mJ..c. Pf1.rv., -6, Nw- YMR.

[34]

KIEFER, J.C. On ~ge. de.vJ..atJ..ol1-6 06 .the. e.mpJ..Jr.J..c.al d.6. 06 ve.c;tof1. c.hanc.e. vaJr.J..abtrv., and a taw 06 J...te.f1.ate.d tog~hm. Pac.J..6J..c. J. Math. (7967), ~, 649-660.

[35]

KIEFER, J.C., WOLFOWITZ, J. On the. de.vJ..atJ..ol1-6 06 the. e.mpJ..Jr.J..c. dJ..-6.tJr.J..buUon 6unc.:Uon 06 ve.c;tof1. c.hanc.e. vaJr.J..abtrv.,. Tf1.a11-6. Amvc. MCLth. Soc.. (7958), §2, p. 773-786.

109

[36]

KOLMOGOROV, A.N., TIKHOMIROV, V.M. E-~nthopy and E-~apac{ty 06 ~~ ~n 6UYlwonai ~pa~eA. AmeA. Math. So~. T!W.~t. (SeA. 2) (7967), }2,

277-364. [37]

KOMLOS, J., MAJOR, P., TUSNAVY, G. An appJWumailon 06 pCVttiai ~UJ11f.> 06 J.nd~p~nd~n-t RV'~ and .th~ ~ampt~ VF. 1. Z. WaM~hun. V~JrW. G~b.

( 7975), [38]

l!,

p. 71 7- 737 •

(7983). A f1.~ma.JtQ on ~mp~~ai m~MWleA. In A FeAMd'LU;\.t 6011. E!1.J.~h L. L~hmann J.n Honof1. 06 hL6 SJ.x.ty-H6.th BJ.f1..thday (7983), 305-327

LE CAM, L.

Wa~wof1..th, Be£mon-t,

[39]

C~60f1.nJ.a.

MAJOR, P. Th~ appJWumailon 06 pCVttiai ~UJ11f.> 06 J.nd~p~nd~n-t RV'~. Z. WaM~hun. v~.

G~b.

(7976),12,273-220.

[40]

MASSART, P. VJ..teA~~ d~ ~onveAg~n~~ ~ t~ ~L(Of1.(m~ d~ ta ~mJ..t~ ~~JU:f1.ai~ POWl t~ pf1.0~eA~M ~mp~qu~. Thu~ d~ 3~ ~yct~ nO 3545 d~ t'UnJ.v~U( d~ P~-Sud (7983).

[ 47 ]

MASSART, P.

VJ..teA~eA

d~ ~onveAg~n~~

deA pf1.0~eA~M ~mp~queA. S(!1.J.~ I, 937-940.

d~ t~ .th(of1.(m~

No.t~ aux

~~JU:f1.ai ~mJ..t~ pOWl C. R. A. S., .t. 296 (20 j~n 7983)

[42]

MASSART, P. RateA 06 ~onV~f1.9~n~~ J.n .th~ ~~nthai limU .th~OI1.em tJOI1. emp~~ai pf1.0~eA~eA (Apf1.J.,t 7985),~ubmJ..t.t~d .to Ann. I~.t. H~~[ PoJ.n~a.Jt~.

[43]

PHILIPP, W. Atmo~.t ~Wl~ J.nva!1.J.an~~ p!1.J.n~pteA va!1.J.abteA. L~UWl~ No.teA ~n Math~mailCJ.,

[44]

POLLARD, V. A c~JU:f1.ai ~mJ..t.th~OI1.~m 6011. ~mp~~ai pJWCeA~eA. Math. So~. S~f1.. A (7982), 235-248.

[45]

POLLARV, V. RateA 06 ~thong unJ.60f1.m ~onv~f1.g~n~~

[46]

SERFLING, R.J. PJWbab~y J.n~qu~eA 60f1. .th~ ~um J.n ~amp~ng wUhou.t f1.~pta~~m~n-t. Ann. S.tat. (7974), vot. 2, nO 7, 39- 48.

[47]

SION, M. On lLYU60~zailon 06 ~~ J.n .topotogJ.cai ~pa~eA. T!W.~. AmeA. Math. So~. (7960), 237-245.

[48]

SKOROHOV, A. V. Th~of1.Y Pf1.ob. Aprt.

[49]

STRASSEN, V. Th~ ~x{J.,.t~nc~ 06 pf1.obab~y m~MWleA wJ..th 9,[v~n ma.Jt9~naL~. Ann. Math. S.ta.t. (7965), ~, 423-439.

[50]

TUSNAVY,

[ 57]

VAPNIK, V.N. CERVONENKIS, A.Y. On .th~ unJ.60f1.m ~onV~f1.g~n~~ 06 f1.~V~ 6f1.~qu~n~eA 06 ~v~nM .to .theiA pf1.obab~eA. Th~of1.. PJWb. Appt.

60f1. ~UJ11f.> 06 B-vaiu~d f1.andom 709, 771-793.

ii,

J. AM~an

(7982). Pf1.~p!1.J.n-t.

J..£,

(7976),

Q,

628-632.

G. A f1.~ma.JtQ on .th~ appf1.oumailon 06 .th~ ~ampt~ VF ~n .th~ mut.tJ.~m~~,[onai ~~. P~!1.J.o~~a Math. Hung. (7977), 53-55.

!'

.

~

(7971), ]£,264-28 • [52]

YURINSKII, V. V. A ~mooiling J.n~quaLUy 6011. eAtimateA 06 .th~ Le. v y-Pf1.ohoJWv ~.tan~~. Th~of1.Y Pf1.ob. Appt. (7975), !Q, 7-70.

[53]

YURINSKII, V.V. On .th~ eAf1.Of1. 06 .th~ gaM~~an appf1.oxJ.mation 60f1. ~onvotu.t,[o~. Th~OI1.. PJWb. App!. (7977), ~, 236-247.

[54]

YUKICH, J. E. UnJ.60f1.m ~xpon~ntiai boun~ 6011. .th~ nof1.m~z~d empJ.f1.J.~ai pf1.0~eA~

I 7985) •

Pf1.~p!1.J.n-t.

MEAN SQUARE CONVERGENCE OF WEAK MARTINGALES

Mariola B. Schwarz Institut

fur Mathematische Statistik

Universitat Gottingen Lotzestr. 13, D-3400 Gottingen

[4J

In

it was shown that the mean square convergence of vector-valued martin-

gales in spaces of Rademacher type or cotype 2 is closely related to the fOllowing property of a Banach space valued martingale Radon measure space

Yf

on

(B,6)

the Borel

(B

f = (f ) : there exists a Gaussian n

cr-algebra of subsets of the Banach

B) such that

(*)

Ilx*fl!;=J

Ix*(x)1

2

B

Y (dx) f

x * E B*

for every

We give a characterization of the class of martingales satisfying not containing

in 00

"n

I: IIS(e*f)11 en' where n=l n 2 is the standard square function.

(e )

for spaces

(e ) in means of

uniformly and having an unconditional basis

convergence of the series

(*) n

is the dual basis and

Furthermore we give necessary (resp. sufficient) conditions for the

L -convergence

2

2 (resp. cotype 2).

of martingales in spaces of type

2

These conditions characterize Banach spaces of Rademacher type or cotype Throughout, Borel

B

A family

[f

n

*

denotes a separable Banach space,

cr-algebra of subsets of

[J n

respect to the filtration J

and

n

Band

(O,J,P)

B

the dual space,

B

the

a probability space.

B-valued random variables forms a weak martingale with

1 n E N} of

Pettis integrable On

S

\ n

E N}

if for evecy

n

E

N

for

(Pettis-)

A

B-valued weak martingale

[f 1 n E N}

M

such that the inequality

1 / f 1 ,; M Ilx"11

f

n

J

is

n

measurable,

k,; n (see e. g.

[ 3 J) •

is uniformly bounded if there is a constant

n

hOlds a.e. for every

x* E B*

and every

n EN.

Note that i f

x* f

=

[x* f n

f = [f 1 n

n

In

E N}

In the following

LEMMA 1. Let

is a

B-valued (weak) martingale then

is a real martingale for each

f = [f

with difference sequence (en)nE N

basis. Assume that

E N}

B

n

1 n E N} [d

n

is a

* x

E

*

x f=

~':

B

B-valued uniformly bounded weak martingale

1 n E N} •

be an unconditional basis in does not contain

in 00

Then the following conditions are equivalent

B

uniformly.

and

(e~)nE N

the dual

111

I:IIS(e* f)11

e is convergent in B 2 n b) there exists a Gaussian measure Y on (B,B) f 2 Ilx*fll~ = Ix*(x)1 Yf(dx) • a) the series

n

such that for each

x

*

E

*

B

J

B

x* f

Proof.

= (x* f )

is for each

n

Thus the square function

S

*

Let the functional

x

* E B*

*

of

x f

x* d

a real martingale with increments

n

is given by

X B* - R be given by

T : B

* * * x ,y E B T

*

is well defined everywhere in the Cartesian product

B

X B*

since

f

is of

* *

weak second order. Furthermore, T is positive (i.e., T(x ,x ) ~ 0 for every * * d ° (0 * * * * * * E B* ). x E B) an symmetr~c ~.e., T(x ,y) T(y ,x) for all X,y Since

f

is uniformly bounded we have

T(x*,/) ,;; (E I: x*(\)2)1/2 (E L: /(\)2)1/2 k

k

* 2 IIS(y* 011 2 Ilx* fl1 II y * fll 2 2

IIS(x 011 =

2 ,;; M 11x*1I II/II which means that

T

We can consider

is bounded.

T

* B

as a map of

e.g. [lJ), which states that an element the natural embedding of continuous in the Assume that

B

B-

into

*

B-topology on

** B • Using a theorem of Banach (see

into

B

u

of

belongs to the image of

i f the functional

, we show that

*

TB

B

in

u(v), v E B* , is is contained in

B.

a) is satisfied. Since

I: II S(e*f) 11 e = I: (E(I:(e*(\))2)1/2 e 2 n n n n k n n

L: (Te*(e*))1/2 e n n n

the series

B*

is convergent in

n

B

n

By theorem 2.1. in B ,

[2J,

T

is a covariance operator of a Gaussian measure

Y

On

i.e., T/(X*) =

J

2

y(dx)

B

Since Ilx* f,1122

Ix*(x)1 2

= II S(x* f)1 12 = Tx* (x* )

the implication Conversely, if

a) => b)

follows.

b) is satisfied, the series e

is convergent in

n

= I: n

B

as

(Te*(e*))1/2 e n

T

n

n

Is a covariance operator of

Yf •

o

112

ln

We recall that a Banach space does not contain Rademacher cotype r < '"

cotype

r < '" • Banach spaces of Rademacher type 2 are of some Rademacher

r 5J).

(see, e.g.,

Using Theorem 5.3. and 5.4. of THEOREM 1. Let

uniformly if it is of certain

'"

(en)nE N

[4J , Lemma 1 and the above remark we get

be an unconditional basis in

*

Band

(en)nE N

the dual

basis. Then i)

B

is of cotype 2 if and only if for every

* 2 en 2.: IIS(enOli

series

in

B

f

the convergence of the 2 is a sufficient condition for the L convergence

n

of

f.

B

ii)

is of type

2.: IIS(e * 01\ 2 e n

n

in

n

2 if and only if for every

the convergence of the series 2 is a necessary condition for the L convergence of f . 0

B

f

COROLLARY 1. In Hilbert space (and only in this space) the convergence of the series

2.: IIS(e* f)11

e is a necessary and sufficient condition for a martingale 2 n n n converge in the L2-nonn. Let

'Tr"y

be the family of all

exists a Gaussian measure

2

x

*

(B,B)

THEOREM 2. Let

for which there

such that

* [2 J

(en)nE N

p > 0 • Assume that

and the techniques of the proof of Lemma

be an unconditional basis in B

does not contain

n

1",

such that for each c 1 (p)

n

o

E B

Using Theorem 2.2. in

and

on

f

(f )

to

rB Ix*(x)1 2 yf(dx)

IIx*f1 12 = for each

Y

f

B-valued weak martingales

f

112.:11 s(e» n

11

2

enl lP

~

J

II

xil P

B,

*

(en)nE N

1 we get the dual basis

uniformly. Then there exist f E

~

Y

y(dx)

B

REFERENCES

[lJ

S. BANACH:

i2J

S.A. CHOBANYAN AND V.I. TARIELADZE : Gaussian characterizations of certain Banach spaces. J. Mult. Anal. 7, 1977.

[3J

K. MUSIAL

r 4J [5J

Theorie des operations lineaires, Warszawa 1932.

Martingales of Pettis integrable functions. Lecture Notes in Math. 794, 1979.

NGUYEN DUY TIEN : On Kolmogorov' s three series theorem and mean square convergence of martingales in Banach spaces. Theor. Prob. Appl. 24(2), 1979. W.A. WOYCZYNSKI : Geometry and martingales in Banach spaces. Advances in Prob., Dekker, 1978.

METRIC ENTROPY AND THE CENTRAL LIMIT THEOREM IN BANACH SPACES J. E. Yukich * Institut de Recherche Mathematique Avancee Universite Louis Pasteur 7 rue Rene Descartes 67084 Strasbourg, France

§l.

INTRODUCTION

The intent of this article is to study the relationship between (i) the central limit theorem in Banach spaces, (ii) the Donsker property for unbounded classes of functions, especially subsets of the Banach dual, and (iii) metric entropy with LP bracketing, p;::.l Before exposing the main results, let us first Set in Dlace the framework for empirical processes, the setting of this paper. Throughout, take (A,A,P) to be a probability space and xi' i;::'l, the coordioo nateS for the countable product (Aoo,Aoo,p ) of copies of (A,A,P). The nth empirical measure for P is defined as P (B)

n

n

I

j=l

Given a class FF(x)

n

-1

Fe

1

, BE A • {x.~B}

J

L2(A,A,P) of real-valued functions with envelope

sup I f(x) I f
valued functions on let

let F;

S:=too(F) equip

S

be the space of all bounded realwith the sup norm, i.e., for

s

I;

S

II sll:= sup Is(f)1

fEF

We note that (S, II II) is a Banach space, non-separable for F infinite. When FF is finite P a.e. the function-indexed emnirical process V

n

(f) (w):= n 1/2

I

is a random vector v n with values in S. Recall [7,8J that F is a P-Donsker class if n converges in law in S to a Gaussian proceSS Gp , indexed by F, with a.s. bounded, 0p-uniformly continuous sample paths (abbreviated BUC). Here,

v

o~f,g) *Current

:=

I

(f_g)2 dP - ([(f- g )dP)2

address: Lehigh University,Math.,Bethlehem,Penn.18015-USA

114

Gp

necessarily has mean 0 and the same covariance as

f fg

cov(Gp(f), Gp(g)) =

dP -

ff

dP

I

v

n

g dP .

Clearly, a necessary condition for F to be P-Donsker is that F be GpBUC, i.e., that the limiting Gaussian proceSS Gp can be chosen to be BUC. Define the mapping h(x)(f)

f (x)

-

f

h: A->- 5

by setting

f dP

for each x ~ A Defining the random variables Xj = h(x j ) , Dudley has shown [6J that if F is GpBUC then the p-Donsker property for F means that Xl satisfies the central limit theorem in (5, II II) , abbreviated Xl E CLT (Recall that if X is a ranoom variable with values in a Banach space B and if (Xn)neN is a sequence of independent copies of X , then X satisfies the CLT if n

L(

I

i=l

X. / /i1) 1

converges weakly to a Radon measure on

B.

5ee

[12,16J

for an exposi-

tion of the CLT in Banach spaces.) In this way we see that central limit theorems for the empirical process v n (f) , f e F , can be viewed as central limit theorems with respect to the norm II lion the Banach space 5. Throughout this paper let (B, II II ) denote a not necessarily separable Banach space , B* the dual space, and B*1 the uni t ball of B* Now a subset H of B* is ca 11 ed a norminf' subset (see [7 J) 1 for B i f and only i f II X II = sup I h (X) I for all x (' B Clearly, hE'H by the Hahn-Banach theorem, B~ is always a norming subset, a fact which We shall exploit in the second section. In this way, limit theorems in B can be viewed as limit theorems for empirical measures on B, uniformly Over a class of functions, such as Bl* ' since for feB * and x(l), ... ,x(n) e R (Ox(l)+"

.+ox(n))(f)

=

f(x(l)+ ... +x(n)) .

In particular, We note that X (with L(X) = P) satisfies the CLT in if and only if B* is P-Donsker. 1

B

115

The relationship between limit theorems for empirical processes and limit theorems for Banach space valued random variables has been studied by several authors. For example, Dudley has shown [7] that the Jain-Marcus CLT for C(S) -valued ranc10m variables [13] is in fact a consequence of Pollard's CLT [17] for empirical processes. Dudley has also shown [7] that the Fortet-l'-Iourier strong law of large numbers [IS] in separable Banach spaces is a consequence of the DeHardt-Dudley law of large numbers for a class 0 f functions [3,7]; We will return to this implication later on and will also provide a short and simpler approach. Limit theorems for vn(f) often involve a metric entropy condition on the index Set F In this article ~e will consider metric entropy wi th

LP

bracketing, defined as

Definition 1.

Given

such that

~

and

h (x)

J(f:J -C)J P P L

such that

[7]:

f,g : A-+ lR, define the bracket [f,g] := {h:A-+ lR

Fe LP (A,A,P)

[f~,f;] wi th

f(x)

.f'ollo~s

~g(x)

Let \'f "F

VXt'

A}

and

N CP) [ ] ..= N(P) [I

3 l~j~m

¢

otherwise.

(~~" F P)

such that

Now log Nfpj (E)

. { m: : = mIn

p~l

Let

3 [f-l' f+] 1

f. [fj,fj]

, E> 0 , ...

and

is referred to as metric entropy

bracketing.

We recall that metric entropy with Ll bracketing has heen especially useful in the study of empirical proceSSes; See e.g. the works of DeHardt [3], Dudley [4,7], Dudley and Philipp [8], Alexander [1], Borisov [2], Gine and Zinn [11], pyke [18], KOlcinskii [14], and Yukich [20,22, 23]. See also the recent monograph by Gaenssler [10]. In particular Nflj

Vn(f) [4,7,8]. One of the main ideas running through this article is that Ll bracketing is not a natural condition for descrihing weak convergence; L2 bracketing is a more natural choice. On the other hand L l hracketing is a more natural condition for descrihing laws of large numhers. These points will be discussed jn the following sections. has

has been used to descrihe the weak convergence o.f'

Finally, We will also USe metric entropy without bracketing which enjoyed wider use, and which is defined as follows:

Definition 2. fli F

~ l~j~m

Let with

N(P) (E,F,P)

;=

min{m:3fl, ... ,fm such that for all

JU-fj)PdP<E P }

ferred to as the metric entropy of

F

Now

10gN(P) (E,F,P)

is re-

116

Having set in place the framework for this paper, We now proceed to the main results, a summary of \,"hieh was announced in [2lJ. Section two discusses the relationsllip between (i) and (iii) of tIle first paragraph, section three studies the relationship between (ii) and (iii), and section four considers metric entropy with L2 bracketing.

§

2.

~lETRI C

ENTROPY A!\'D TIlL eLT FOR TIlE U\IT BALL 01' THE BANACH DUAL

Our starting point is the following result of Dudley [7J, which adds to Mourier's well known law of large numhers [15J.

Theorem 1. (cL section 6.1 of [7]1 Let X, Xl"" he a sequence of i.i.d. random variables with values in a separable Banach spaCe (B, II II). The following are equivalent: (i ) X sat i s fi est hest ron g 1 aw 0 f 1 a r g e n urnbe r 5 (S LDI) , (ii) E IIXII < and 00,

(i i i) N (1)

(E, B1* ,P) <

[ J

00

\J E > 0 .

I t is reasonable to ask \,-hether analogous results hold for those

satisfying the eLT The following theorems answer this query in the affirmative, showing that the study of the CLT in separahle

X

Hilbert spaceS

H

is conveniently studied through the USe of

N(2)

* , where III* [ J (E,Hl,P)

is the unit 11 all or the Hi 11,ert dual.

In fact, in the same way that the equivalence of XECLT

and

E{X}

o,

E II xii 2 <

00

characterizes, modulo an isomorpllism, separable Hilbert spaces, the following results show that the same is true for the equivalence of the condi ti ons

B*

1

is P-Donsker and inf N(2) 00 [J

( E,

Bl < * ' P) .

00

P

centered.

117

This is contained in the following results.

Theorem 2. Let X be a random variahle with valueS in (B, 1111), which need not be separable; L(X) = P Then for all P':' 1 the following are equivalent: (i)

(ii)

E

IlxII P

inf

< 00

N(P)

P

<

[J

pO

If

, and 00

is tight then the following are equivalent: (iii)

EllxllP

(iv)

N(P)

[ J

and

<00,

(E,Bl*,P)

<00

t'E>O.

Remarks Using the equivalence of (iii) and (iv) with P 1 We directly obtain the double implication (ii)~(iiii of Theorem 1 without using the SLL~ property of X . (1)

As is well known [12,16J the moment condition EIIXl1 2 <00 is in general neither necessary nor sufficient for X IC CLT ; it may also be (2)

easily Seen that the entropy conAition

.

f

N(P)

~~O'[ J

(

F)

E"P

<00

.

IS

.

neIther

necessary nor sufficient for F to be P-Donsker. The interest of Theorem 2 stems from the fact that when F is B~ these are in fact equivalent conditions and they consequently share the same properties.

The next result is essentially a consequence of Theorem 2. Theorem 3. Let X be a centered random variable with values in a Banach space (B, II II) ; LeX) = ? Consider the following properties of P: (i)

X

(ii)

B*

IC

CLT

1

is a P-Donsker class of functions,

(i i i) E Ilxll 2

< 00

,

* (E,Bl,P)

(iv)

N (2)

(vi

* inf N (2) (E,Bl,P) [ J E>O

[ J

'rIE > 0 ,

< 00

< 00

We have: (a)

If

P

is tight and

all equivalent.

B

is a Hilbert space then (i)-(v) are

118

(b)

If B is separable, then the equivalence of (i), (ii), (iii), and (v) is equivalent to the fact that B is isomorphic to a Hilbert space.

For the proof, We need only observe that when B is a separable Hilbert space, the equivalence of (i), (ii) and (iii) stems from the introductory remarks. For separable type 2 Banach spaces B we have sharp metric entropy conditions insuring the P-Donsker property for B*1 Theorem 4. Let a centered law.

(B, II II) If

be a separable type 2 Banach space and

P

inf E>O

then

B*

is a P-Donsker class. Conversely, if P is a law on any separable Banach space B and if Bi is P-Donsker, then for all 1

p < 2 ,

The proof of Theorem 4 follows at once from Theorem 2. For the first part We need only USe implication (ii)~ (i) with p = 2 ; for the second We USe implication (iii)~ (iv) with p < 2 together with the f'lct that i f XECLT then EllxllP
N[Pj

=

-

+

brackets [f.,f.] J J with

We will first show (ii)

(E,B * l , P). , j

=

~

(i).

Let

E> 0

and

By definition there is a collection of

l, ... ,m, such that given any

3l
+

fk(x) < f(x) < fk(x)

Vx "

B

and

Now B*1 is a norming subset of B* and thus for fixed x EO B We See that f(x) runs OVer all of the values between -llxll and Ilxll as This implies that for any fixed x , the sum of the f runs over B*1

119

m

I

differences of the brackets

j =1

(f:(x)-f~(x)) J

must be at least as large

J

Indeed, suppose this Here not the caSe.

as 211xll that

Then:3

X

o EB

such

Thus there exists an -llxoll < a < Ilxoll and a 0> 0 such that the open interval (a- 0 , a+o) is disj 0 int from the union of closed intervals m _ + .U [f.(xO),f.(xO)J J=1 J J

an

fEBl*

Moreover, by the Hahn-Banach theorem, there exists

such that

there exists

1< j.:::.m

This leads to a contradiction since

a

=

such that

+

fj (x) .:::. f(x) .:::. f j (x)

VX

E

R •

Thus, we have shown that m

For all

I

<

IIXII

j=l P > 1

f:(X)

-

fj (X)

J

a.s.

there exists a constant P ( (f +. - f.) (X))

J

J

Integrating this with respect to N[Pj

(s,B~,P)

completing

C . = C (p, m)

such

a.s. P

and using the definition of

shoHs that

(ii)~(i)

To show (i)=)(ii) it suffices to consider the single bracket defined by

fi (x) = -llxll

and

f~ (x)

= Ilxll

and to take

f

s = 2 Ilxll PdP (x) .

If P is a tight measure then We may show the stronger imnlication (iii)=9(iv); our proof is inspired by the proof of Proposition 6.1. 7 of [7J. Let p> 1 be fixed and note that for all s > 0 there is a compact Set K~ B such that

fBIK

IlxIIPdP<sP/4

The elements of

Bi ' restricted to

K, form a uniformly bounded

120

equicontinuous family ano hence this family is totally bounded ~or the sup norm II ilK on K by the ArzeIa-Ascoli theorem. ~ B* ' m < "" , such that Iff ~ B}* l Ilf - f. II J

for some g.

J+m

Let

J

=f.+E/4

On

J

N~Pj

(E

f , ... , f m l

< E/4

K

Then for every

Thus,

Take

g.=f.-E/4 J

K,

fE B* 1

B~,P)

J

on

K , g. (x) = - II xii

gj+m(X) = Ilxll if

J

for

Ilf - f j Ii K < E/4

< 2m < 00 , proving

x ¢K

,

then

(iii)~ (iv)

~or

for

x ¢K ,

all

j = 1, ... ,m

g.
-

J +m

and completing the Q.E.D.

proof of Theorem 2.

§3.

and

THE P-DONSKER PROPERTY AND METRIC ENTROPY is often an

The results of the above section show that appropriate tool for describing the P-Donsker property above results also suggest that

~or

the

N~P~ (E,F,P) , p f 2 , will not in

general be a satisfactory tool to describe the P-Donsker property for unbounded classes F S LP(A,A,P) . This is indeed the caSe: the following propositions show even in the preSence of an envelope condition on F, that (E,F,P) , P 2 , can not possibly give sharp resul ts. This may be seen by cons idering suitably chosen subsets of the dual to (1'.00, II 11 00 ), the Banach space of bounded functions on N+ equipped with the sup norm. More precisely, these subsets are classes of functions of the form

+

NtPj

121

F

:=

a,S

0:.

a. s . lA : s . J J

J

.

J

J

=

n,

or

0

where A. , j ~ 1 , is a sequence of disjoint subsets of A with J p. - P(A.)=j-S for some S> 1 and a. = J.a for some o < a < S - 1 J J J Since elements of F define measureS on /N+ , we may view Fa, S as a,S a subset of the dual to (tOO, II 1100)

We will consider the cases

p = 1 , P > Z and 1 < p < Z in this order; much of what follows may be found in [19J. Our f'irst proposition shows that a theorem of Dudley, which is recalled below, is far from being the "best possible". Theorem (cf. Theorem 3.1 of [5J). Suppose that F has envelope FF€ LP(A,A,PI for some p> Z Suppose that there exists y , o < '( < 1 - Z/ P an d M < such that 00

for

E

small enough.

is a P-Donsker class.

Z< p < 4

For all

Proposi tion-l.

F

Then

there exist

Donsker classes

P-

F

with

-y

FF

€

LP(A,A,P)

(E,F,P) > ZE

and

where

is any number leSS

y

than liZ. Fix

Proof. p/(p-Z) above.

and

Z