A weak convergence approach to the theory of large deviations

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Author: Paul Dupuis | Richard S. Ellis

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viii CONTENTS do not depend smoothly on the spatial parameter. As a result of this nonsmooth dependence, classical methods for studying these processes are not applicable. An example in which the weak convergence approach is particularly useful is given in Chapter 7 of the book, where we study a random walk model with discontinuous statistics. Markov processes with continuous statistics will also be analyzed. These processes are characterized by the fact that the components of their generators depend smoothly on the spatial parameter. In the cases of both continuous statistics and discontinuous statistics, the weak convergence approach enables us not only to reproduce known results, often under weaker conditions, but also to treat new problems. The theory of large deviations has inspired a variety of approaches different from the one presented here.Book-length treatments that develop some of these approaches and apply them to specific problems include the texts by Dembo and Zeitouni [28], Deuschel and Stroock [29], Ellis [45], Freidlin and Wentzell [54], Shwartz and Weiss [83], Varadhan [88], and Wentzell [93] We would like to thank a number of people for their help. Christopher Boucher, Michelle Boué, Gordon Kieffer, and Kavita Ramanan each read parts of the manuscript closely and suggested improvements. Andrew Barron, Alejandro de Acosta, Amir Dembo, Wendell Fleming, Joseph Horowitz, Steven Shreve, and Srinivasa Varadhan made useful comments that were incorporated into the book. We gratefully acknowledge the support of the National Science Foundation and thank the Division of Applied Mathematics at Brown University and the Department of Mathematics and Statistics at the University of Massachusetts for providing excellent working environments. One of us (P.D.) also thanks the Army Research Office for its support. Our greatest debt of gratitude is owed to Ofer Zeitouni, whose many insightful remarks significantly affected the final form of this book.

Conventions We adopt the following conventions throughout the book. • Subsequences are labeled by the same index as the original sequence. • The σ-field on a Polish space is always taken to be the Borel σ-field. • By a probability measure on a Polish space we mean a Borel probability measure. • A function mapping a Polish space into a Polish space is called measurable if it is measurable with respect to the underlying Borel σ-fields. • Since all control problems that we consider are optimal control problems, we will drop the term "optimal."

• The usual rules for arithmetic involving ∞ are adopted; e.g., 0 • ∞ = 0. • We abbreviate "almost surely" as "a.s." and "with probability 1" as "w.p.1."

For historical background on large deviations the reader is referred to the texts of Dembo and Zeitouni and of Deuschel and Stroock.

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E I N7OQPSRUTWVXTYOQP[Z]\,+]\Y`U\ b r zxzd{ lmb 4gJf e okcYoqgJfz 6 B$CB$ 6 B B% 6 BPJB r f e 6 BPJB$ b rHt zXg r zxzd{ lmb ckj r co } r zXbTS{;bdfhpxbg~} gJnxb t zd{ v zXbdckz Á 0Rjk l& g~}; z r cYozW|bxzÁ ½ Á r z d j;bdnbÁ z r cYozW|bxz "b Á d q L ckj;bdf - 9A< <7= °Wb! 0 Á d q L { f;o }gJn l t w}gJn okf§pxgJl!y r pdc9zd{ v zXbdcWz g~} Z / KK ,; 'V3L5F783J3L5' + / < ?%'.>(' ª 90+*) ,+ 1 9 3 9 ,+ 6 BKJB_' a/F78- ; ,B V 7@>h3 b ? d @PW ' << 7 6 BKY2B K > e ' ; 365F783 9 P "b Á d q L 3652' +SP >U7@-0- j °Wb 0 Á d q L 7 + 1[3L5C/ ; - 9A< ° b! 0 Á d LR9 P W 7 < = Á ½ Á B ) ,+ 1 9 3 9 @+ 2B B ; ,< ' T 5F783 ; 3L>L' +2Z 365' +; 3L5 90; 90 <9 3HG,B K - 365 / Z 5 b d ' C/ 90KK '.1 TU9 3L5 365't 3 @K - ,Z G 90;*+ ' 9 3L5'.>*7 < 'D3L> 9 7@?-A' ;]K 7@).' + >V7 o ; ' K 7@>L7@?-0' ,+ ' a 9 3 9A; 7,+o78/ ; 1 >.-_3 @K - ,Z,9 )7@- ;LK 78).' 7 + 1 9 3Q)D'.>]367 90+ -AG < 7 ] ' ; ; ' +2; 'U3 ) ,+;]9 1'D> 7 W 7 K -I7@)D'/ KK '.>? / + 1S7 + 1 - T 'D>? / + 1 9A+ 3L5 90; ) ,+ 36'^gC3.BX 5 ' + T ; 3N783L' 3L5' ; '=? / + 1 ;P > < K 9 9 < ; ; 0 9 + , K , Z U T 9 3652'=' > )7@- '7 />]' 365'mt 3 - G,B `5'DG[5 -A1 365V365' ; 7 < ' P / + )^3 9 @+ k 7 ;9A+ 3L5' ) >L>L' ;LK ,+ 1 9A+Z W 7 K -I78).' K > 90+ ) 90K -A' ? / + 1 ;Q9A+/) 5F7 K 3L'.> 6 E `52' >]' < 6 BPJB I BUX 5 'U78- ; K > e 'U365F7O3 k 57 ; ) ,

%

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j r zpxgJl!y r pdc t bdHb t zXbdcWz®okfckj;b t c~gxy;g t gT/Hw SgJnb r pxj pxgJl!y r pdczd{ v zXbdc : g~} bj r Hb(ckj;b®{ f;o }gJndl {1yMy;bdn v gJ{ f e - ,ZBA 'rg K 3 b C 8 d I 9- A< ; / K ; / K E L@ C D C 90F +2P· º k b" d 3 b" d p > @ ? 8<7= > b ) d -UgJnb r pxjpxgJl!y r pdczd{ v zXbd c : g~;} bj r Hb#cWj;b®{ f;o }gJn l t g bdn v gJ{ f e ,ZMA K b 90+2P b" b" - 90< 9A+2P 9A+2P 8<7= > ? @ - > '^g E C @ 3 C 8 d I N C F · º k d 3 d p + >L1'D>U3 \K > e 'm3652' W 7 K -I7@)D'? / + 1 ; T ' 90+ 36> 1/2).' 8 b! d q p C @ - ,ZBA > 'rg K E LC @ 3 b C 8 d I 7 + 1[/ ; 'm3L5' ; 7 < 'm>L' K >L' ; ' + 36783 9 ,+[P > < /-07E7 ;90+1) 5F7 K 36'D> 6 2+ 7 < '.-AG 8 b d q 8 b d q p 90+ A P A a > @ 8 < = : ¬ _ 8 b # a 8 0 C a 8 d ¼ W° b a 8 0# d 3 b C a 8 d ª K ;]9 1' P > ,< 7 + 7 KK > g 90< 783 9 ,+ 78> Z / < ' + 3 365' K > @P ; 7@>]' ' ;]; ' + 3 9 7@-A-AG\3652' ; 7 < '7 ; b7 d k b? d -

p

63 5' K > 8P ; @P 3L5' W 7 K -I7@)D' K > 90+ ) 90K -A'? / + 1 ;9A+ 3652' T '7 ] 3 ,K - ,Z G,B_`52' ,+ - G 1 9 -i'.>]' + )D' + T 90; 365F7O3 3 90;=+ 3J) ,+ 3 9A+ / / ;TU9 365X>]' ;]K 'D)D3E3 365 9A; 3 ,K - ,Z G@BS`52' < 7 90+M+ ' T 90+2Z >L'D1 9 ' + 3 TU9 -A- ?%'3 ; 5 T 365783 T B K B% C a q C a 9A<K - 9 ' ; 3 b C a ,d % 3 b C a d T 5' + 3L5' ; ' C/2' + )D' @P >]/ ++9A+Z ) ,; 3 ;9A; ? / + 1'D1 BQ`5 90;U9A<K - 9 ).783 9,+9A; 7\) ,+; ' a/' + ).' 8PXW ' @P @P 3L5' W 7 K -I7@)D' K > 90+ ) 90K -0':? / + 1 ;S90+ 365' T '7 ] 3 @K - ,Z G o9 3 TU9 -0- ?%' + 'D).' ;]; 7@>hG*3 ) ,+;]9 1'D> ) @+ae '.> Z ' + 3 ; ' a/' + )D' ; @P 7@1 - < '.7 ; /2>L' ; 7 + 1_) @+ 3L> -A-0'D1 ' 9 ).7@- < '7 ; />L' ; 3 ,Z 'D3L5'.> TU9 3L5=3652' 9 > - 9A<9 3 ; T 52' +E; ' a/' + )D' @P >]/ ++9A+Z ) @; 3 ; 90; ? / + 1'.1 B `5' P -0- TU9A+Z -A' @P ; 8P 365' W 7 K -I7@)D' K > 90+ ) 90K -A'=? / + 1 ; 3 3L5' t 3 ,K - @Z G,B `52' K > @PW90; 7@17 K 36'D1 P > ,< B 9Ae ' + 7 + 7@1 <9A;L;L9 ?-0') ,+ 3L> - ; ' a/' + ).' _ q p.pp @ 2T '=>]'.)78-0- 3L5' 7@1 - < '.7 ; />]'

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&© ¤§¤§£ Z6\,+]\,+&\ b r zxzd{ lmb 4gJf e okcµoqgJf 2B B a bdc b a 0 a 0 a > d v b r yQng v r v o t okcYwziy r pxb r f e tgJbdf c _ r /Hf okeHbdC fa v bnbx4ziy;gJbxfpdzdcµo okHbdb nt w r zXzdbTc~SgX{ pxbdj fhr zdpxbcYoqp g~} bdn fhlb t ozxgJzd3f o v bRpxgJ f;cYn/Hg okHbdlmf b a zd{ r nf bxez r _ zdcigX0 pxj r jYzdcYoqk lp bdn fhf b t 8 @ a e r M r e t t r pxgJf;cYnxg tkt b e bdl!yQokn oqp rHt lmb r zd{ nbxz C a 8 0 @ jYk l& e bW&fhb e gJf ckj oz$yQng v r v o t okcµw±ziy r pxb3zd{ pxjckr j r e c byozd ckJz - 90< zd{ ; v/ zXK bT8S<7{;=bdfh¬ pxbb _ g~8} ¼ C a j8c k l ° d zd{; pxj³d cWj _ c 8 q y _ C C a 8 q C C a f r f b_e b _ 8 d q C C a j;bdfcWj;bdnb r @ r a8 C a r e 8d C a Ca

8

qp

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X

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n > ; 3 ; 5 T 5 @P 3L5'J-0' << 7 9A; 1' e ; ; A 9 + 0- ' 'D- 'D3 63 5't

t

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- 90< 9A+2P

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N 90F +2P k b" d 3 "b d ·º

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JOhO O £¥¦S¨W£ © ¦S¦© )O PSR&\ X5 ' K > e ' b! 90+2P b" b" - 90< 9A+2P 8<7= 8 d N F · º k d 3 d L ?%' 7 ; 9A+5Z '.)^3 9 ,+ 6 B @ T 5'D>L' 3652'U7 + 7@- ,Z / ;WT '.7 ] 3 ,K - ,Z G=/ KK 'D> ? / + 1 90; K > e '.1 B W 'D3 Z,9 e ' + 7 ; T 'D-0- 7 ; 7 + G ; /2? ; ' C/2' + )D' @P @ jk l 78- ,+Z=T 5 9 )N5*3L5'/ +9 P > < -AG\? / + 12'.1 P / + )^3 9 @+; 8 b! d ) ,+e '.> Z '@B \ >s'7@)N5 @ ) ,+;]9 1'D>7 + 781 <9A;L;]9 ?-A' ) ,+ 3L> - ; ' a/' + ).' _ 8 ; 783 90;hP G 9A+Z 8 b d N A a > @ 8 < = : ¬ _ 8 b # a 8 0 C a 8 d ¼ °Wb a 8 0# d 3 b C a 8 d ª 7 + 1X1' + 36'4?G _ 8 7 + 1 C a 8 365') >]>L' ;]K @+ 1 90+2Z 7@1 - < '.7 ; /2>L'*7 + 1 ) @+ 3L> -A-0'D1 ' 9 ).7@- < '.7 ; /2>L'@B Z 9A+ ).' 8 b d D ¼ 3 ¼ = `52' >]' < 6 B$ B 6 )7 + ?i'm7 K2K - 9 '.1 B `52'.>L''rg 90; 3 ; 7 ; /2? ; ' C/2' + )D' @P b_ 8 0 C a 8 d T 5 9 )N5:) ,+e '.> Z ' ;9A+ 1 9A; 36> 9 ?/ 3 9 @+ 3 7 K 7 9 > @PQ; 3 )N5F7 ; 3 9 ) ] 'D> + 'D- ; b _ 0 C a d T 5'.>]' T B K B _ q _ q C a 365a/ ;Sb_ 8 d C C a B D G ) > -A-I7@>hG ) B B 7 + 1 ' +2; ' + ; : + ' C/7@- 9 3HG 8 < : _ 8 b # 8 0 C 8 d W° b 8 0# d a a ¼ a ¬ @ = ª 8< : ¬ A b # d _ 8 b # a 8 0 Ca 8 d ¼ A b # d °Wb a 8 0# d N ¬ b _ 8 ¼bC9a 8 c ° d p q @ = ª ? ? )

&# %

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`5 9A; G 9 '.-01 ;

d

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8 < : ¬ _ 8 b # a 8 0 C a 8 d ¼ °Wb a 8 0# d

= ª

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T 5 9 )N5 A9 + 36/> + 90L'm'^g 90; 3 ; 7\>67 + 1 ,f < e 7@> 9 @7 ?-A' ; /)N5[365F7O37@- ,+2Z 7 ; /? ; 2 ; / ? ' C/2' + D) ' b

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8 0 C9a 8 0 b_ 8 d 0 ¬ b_ 8 ¼ Ca 8 c ° dLd C

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d p

3

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5

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8 b! d 8 < : _ 8 b # 8 0 C 8 d °Wb 8 0# d - 9A< 90+2P A a a a ¼ a 8 @ = ª ¬ - 9A< 90+2P A a ¬ b_ 8 b ¼ C9a 8 c ° d 3 b C9a 8 d 8 - 9A< 90+2P

q

N

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9 ' ;L_ ¬ ¼ : - 90< 9A+2P ¬ b _ 8 ¼ C a 8 c ° d N ¬ b _ ¼ C a c ° d N k b C a d p 8<7=

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P > 3L5'm' + 3 9 >L' ; ' C/2' + )D'

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C

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0 · º 95 0 ·º ¼ 8 ¼=

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8 E

b d '^g D) ' K 3365F7O3 3 90; >]' K 0- 7@).'D1S?G 3 5F7 ; 3L5' ; 7 < '>L' K >L' ; ' + 36783 9,+VP > < /2-I7E7 ; `5' L> ' < 0 9 < 0 9 ; ; ; ; 8 P ; P 6 B B B (m /2)N5S365783 U ' ).-07 63 5F7O33L5'.>]'m'^g 3 7 >L'D- / ? D' 3 > 7@-A- j

I 95

D

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`5' K7 0- 7@).'

W Z

8<7=

9A+ ).' ) ,+e 'D> Z ' + ).' 90+ 6 BKY2B G 9 '.-A1 ; 3652''^g 90; 36' 3652' P -0- TU9A+ZRK > ,K '.>h3 90+29 3 9 7@-%1 90; 36> 9 ?/23 9 ,+ _

K 87 3L3L'.> + 8PUK > @P @ P 3L5' d ; /2)N5S365783 p

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0 0

»

2

0

¼ °Wb! 0# d]d _ b d D k b d d p \ >Q'7@)N5 4j

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L 0 0 ppp 0 @®C

b

! b 0 # d

" !

0 ¬

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6

3 @ Z 'D3L5'.> TU9 3L5_365' K > F ? 7@? 9 - 9 3HG

- 9A<9 3 ;

- 9A<

8<7= Z

_

9A+ ).' P > '.7@)N5

@

8< : ¬

= ª

8 b

! b a 8 0 # d

H

¼ °Wb a 8 0# d

q

¬ b

_ b d c ! b 0 d 0 C a

d q

8

E

¼ °Wb! 0# d]d _ b d d 0 _ 0 7 + 1 b_ 8 d q _ p

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` b 8 b 0 d - @Z 8 b 0 d C 8 b 0 d d W° b 0 d C a 8 b d q b C a 8 c ° d b 0 d j » 8 b 0 d

¬ b ! 8 b! 0 # d ¼ W° b 0 # dLd C a 8 b d C ¬ b C a 8 c ! 8 ¼ C a 8 c ° d q b C a 8 c ° d b 0 d j » 8 b 0 d b ° b b C 8 0 j 8 0 ¬ b _ 8 b¼ C a 8 c ° d p c d d d »

q a `5' -I7 ; 3U- 9A+ ' 90; 7\) @+; ' C/2' + )D' @P K 7@>]3 b P d 8PXW '

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0 ) ¨ =!. 1 9 ; ) $U(' k ) *)M ) @') #') ') $J# 3H*Q.! ) '< '< )8}

* & @ $ 0 ) ¨ =!. 1 ; k 9 ) $U(' ) ) > (0 ¨ . (M MNma;om ' " $ ch') P e & ' ;

* > )* > ^

ilk 31 . 1 ' ) ')

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%#9 !8 ! # .." H ') # ')"3H" $ Y M [ , $%n ." ') $ $ R " $ q'< 9 1'L# ) V8! ') . % D "%4#;) #'<#"#"[ & & ) ') * E'); $ ¨ . (M M ) ilk ) 1; (' ) # m71; 2 M > ) > ^ > M)5 > MM}*%$ Y 5 [ , $;n .!S')! $ $ R " $c# '&% 9 '<.. & ') & 3 ) 8U') 9 1' B # $ i [') .) 8W E 4 8! n 3H ')*Q'< 3H *Q 7 D } x "A3 > MM $ ,- .! 4 M B $ Y 5 >[ , $ > MM $ Y 5 * [ , $Nn .!q') na$#b ') " $ *Q.!#'<# '))* H 8! l X'<?1') 9 '<# 8Wr B ')$< 8 !9f A ; $ i "')7 .) 8W E 4 S8! n 3%')*_"') 3H *_ x A3 > M)M $ & Dv0 Y 5 [ , $%n ." ') &$PO ! $ # ) #') ') :8! * ) 8 #'L 8W!r :8! % r[. ' B # ¤* ')! ) 8y' ') ') $ T ' \n k ) ¨ . \n ) ( ').. ') $ 9 ; ) > MM $ _ Y 55 [ $ $ R " $ l') 9 1'L# ) ?8!) f'^9 #')[')) 8 #'<* r& $U9R0S0 ) ¨ . (M M ) > * ) > ^ > * > M=<5$

Y 5 [ $ $ RS"" $ 0S'>. " 2 ! (.) 1 4 1\«\m (' m7 0 2 2:(0 T ' (' m[2 ' m7n ; $ ]0 h^ > M= $ Y 5 [ $ $ S R "" $ l') 9 1'< £8!) Q *Q. '<y* ')

').."['<# ( *a# '< 1'L - *Q." [');* ')! $S9R0S0 ) ¨

13n k (0 7m n32 $ . 9 ^V@ ')9

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^Ac ^%,$)#D*UH $ T ' \n k ) ¨ . \n ) ( 9 ; ) 5 &) * > ^ )* =)= > M)M3%* $ Y 54M [ $ $ R-"Q') #$Ena$ c 3H $ x}"8! * 1') 9 1'L# ) K. . A3 ) 8£ ^ *_." "') . &H ()8y' ') ') $L9R0S0 ) ¨ . (M M ) > ) > > 5 ^ > > > > M=M%$ Y [ $ ] $ RY S') $ $ P $r$&(. /VL«(¨ . \n313232\132NY 0k (. n ' 1 . m (' m7 0d(0 0 0 «(1 .) 1 0 n31 $ b 8c 3fe ! <]0 h^ > M= $ Y >[ c $ $ *_"9 $ & ![! ) . 2 ')2"A3 ')L).!#"*R')(H ')P#"h) ; & $ & 9R0< ;op " 2\1 1 ' 0 0S'>. #; 18Tg"\2 ' 1 132 .')9 ^ > $ & "#%_ ! ! * & !8! *R'< : * # !& *R'L# % % > M $ & & # *_" ') $ Y * [ c $ $ *Q9 $ R ". 2 ')2"# :')!Q.!*_')£PH ')d 4 & $ 9 ; ) $U(' k ) W <' m ) 5 ) )* M ^ 5 > M =!$ Y [ c $ $$ *_"9'< $ c $ " $ 4 1 ' 1 . m 0 m[2 ' m7n (0 T ' \n k 2 ' m7n W <' m % ; 0 0S'>. #; $ . 9 ^A@ ')9 V " > M $ Y 5 [ c $ $ *_"9_') $ $ $ 0 0S'>. #;a;[1 G$U(. /VL«c¨ . \n313232\132 (0 K m[2\n3\2m ' "dTl#; = p ' m7 0 2 $ . "9 ^A@ 1')9 P]0 h^ > MM $ Y [ b!$ ')q P# $h 1') 9 1'L# ) Q8W * ') ')4R* '< $ ilk ) ¨ . (M M ) 9( ; ) *}* ) <* 5 ^ M > M $ Y [ , $ & 2' * b!$ ) D ') E$ $ !3H*_9'') $ l') 9 1'L# ) o *Q}8!)

*Q. [')O.! & 2 '<2""A3_* ') $ 9R0S0 ) ¨ . (M M ) ) ^ = > M M!$ Y [ E$ $ "" i') $ $ 2 *R'< $ - 0S'>. (= n ' m7 0G' W 1 . (' m7 0 2 132\1 (. n k $ $ ' B " ,-2" "9 *Q. ')43 g]} -^ 8Z " > MM $ Y = [ $?b!$ ') $ 9)1') S. P# 2 '<# ".%# 2) ! ') 3 ')" . & 2! *Q $ +*) 4m J1 . p 1 0S' m ; & =*(' m7 0 2 * &) <* 54= ^ * > M $ Y M [ #f$4na$ & (D ') @ $ $ *Q $ ilk 13 . " `_ 'X. 1 % %; ¨ . (MN;[1 2 * ) 8T 'X= m7132 m 0 $U(' k 1 %(' m7n32 ( 0 - ' 2 9 ;om7n (' m7 0 2 $ ] %B ') # *Q ')* > M M%$ Y [ ] $gb '< $ l') 9 1'L# ) o }2) ! 8! D')# 8W\# ')"y) 8 ') .! &H $ 9R0S0 ) ¨ . (M M ) > = ) > > ^ > M= > M)M $

Y >[ & $ '< #'< ')0') $ R $ $. 0 m (0 $ ' m7 0 (0 T ' \n k 2 ' m7n 0 ;[n = ; = 2 $ . "9 ^ @ 1')9 *]0 ^ > M==!$ Y * [ ^ $ "8 $ l') 9 1'< ?h!3H ')*Q'<P3H *_c'<¯&% '<#"R. % $ i . (0 2 ) 9 1 . ) $&(' k ) Tl\n ) * > ) u^ *L5 > MM $ Y } [ #f$ , $ ')! $ $ $ # r 8W!r '<08! :!EDO . &H ¤D" 4; D "8Z $ ilk ) ¨ . (M M ) 9( ;om7n ) ) |^ > MM3*%$ & !') $

Y 5 [ #f$ , $ &P )' $ $ $ # \#¤8Wr ')U8! ZDU! )>"%4#;) * 1' $ ¨ . (M M ) ilk ) 1; (' 1 # m71; 2 M)5 ) > ^ }} > MM $ Y [ $#b!$ $?9(<. m %(' m7 0%(0 I 1 / 0 0 «(1 .) 1 0 n31 $ 1 ' k 2g_ . (0 ¨ . \n313232\132 m'k ( 9 ;om7n (' m7 0 2 ' T ' \n k 2 ' m7nTg" 2 ' 1 ilk 13 . " $ &A ,- O ')*f2 "9 > M=)5$ Y [ $ b!$ $ I 1 / 0 0 «(1 .) 1 0 n31 $ 1 ' k 2 (0 Tgm 0 (= ; (. ;o"£¨ 1 . '>=?. M1 T ' \n k 2 ' m7n 0 0 p 'X. #; ( 0 # ma; ' 1 . m 0 ¨ . (MN;[1 2 "* )8Tg"\2 ' 1 2 (0 0 0S'X. #; $ " r')q ! P > MM $ Y [ $rb!$ ! f'<d, $ln .! $%J= 1 . m7n ; $ 1 ' k 2_ . T ' \n k 2 ' m7n 0 0S'X. #;q¨ . (MN;[1 2 m 0 0 0S' m 0S= = 2 i m 1 $ . 9 ^V@ ')9 *]0 h^ > M)M3%* $ Y = [ $ b!$ '< $ ^ " $ T ' \n k 2 ' m7n 9<. m %(' m7 0 9 ; . m ' k 2 (0 9 ;om7n (' m7 0 2 $ & . . ') #'L# ) $ Y M [ b!$ 3H ') !b $ ')*R'< $ l') 9 1'< !V8! . &H D" . 4 r CB * 4# $L9R0S0 ) ¨ . (M M ) > ) > |^ * > M)= $ Y [ #f$ ') B ) q 8 $ !9&. &. P# ED 4#')2""A3 )8l 8W [ £ 9)3'): 4# #'< "*_ * 8! &'.! . ') "h & 9/* H E'<& *Q. #'< $ 0 =!0 ) $&(' k ) ¨ k " 2 ) * ) u^ M3* > M $ Y >[ $ L')*_ $ * 9 #'<['<# ) 8c 4 " * *_'< "') 9 '<# $ T ' \n k 2 ' m7n32 L* 5 ) * M ^ )* =<5 > M)==!$ Y * [ #f$ #$ 9! ;" $ q'< 9 1'< 8!) # #'T r& )8%*a# ^;* ') #')!* E'); $ ilk ) ¨ . (M M ) ( 9 ; ) * > ) |^ > > M $ Y [ x $ ) $ c ! % 4#;A35) 8O ^{ !9f^ #')P8! * $ +*) $U(' k ) 9R0< ; ) 9( ; ) ') > = ^ > M > $ Y 5[ , $ } ] 3+')!R $ ] *Q* " $ ') '<" .! &H & ) 9 ') . . ') "*_V *Q $L9R0S0 ) ¨ . (M M ) > ') > ^ M3* > M= $ Y [ , $ } ] 3j') R $ ] *Q* " $ ') ')# . &H & & )-1') 9 1'L# ) $ 9R0S0 ) ¨ . (M M ) > ') M ^ M > M= $ Y [ $ $ * z') c $ @ E''< $ ').')[# ') 9 '<# '<¯ 9) 91'D $ & T ' MN;[ 1 ¨ . \n313232\132 ( 0 1; (' 1 i m7n32 . ')9 5 ^ = H P > MM > $ " r')q ! $ @ "*

* ,- 9 Do,- &) 2 ')2""A3 $

= Y [ $ &$ ; ') $ T ' \n k 2 ' 7m nL4 m J1 . 1 0S' m ; & =P(' m7 0 2NY (0 . 9 ^A@ ')9 V " > M= $ Y = [ #f$#$ ,- ') ; $ j8W!r '<>. "[". )8(') 9

$ ! ') - "& 1 i . 1 0 2 m 0 ¨ . (M MNma;om ' " @ ,^ ;1'< > MM > $ Y M [ $ $ H L'<8 "1') $ 0 0 «(1 9R0< ;o"\2m[2 $ ,- ! & 0 x

- 0S'X. ( = n ' m7 0 m ' k 9 o; m7n ( ' m7 0 2 $

Y =M [ #f$na$ c 4 $ 9 "*_a . &H & $ ilk ) ¨ . (M M ) 9 ; ) Y M [ #f$na$ c 4 $ 9 "*_a . &H & & $ ilk ) ¨ . (M M ) 9( ; )

1'< !c8!) '< &% '<#"

1 'L# ) $ & @ $ ' )' ( 0 T ' (' m[2 ' m73n 2 . )' 9 > M= ^ * > =!$

A3o,- ,- " > M $ Y = [ $ $ $ - S0 '>. (= n ' 7m h 0 ' T ' \n k 2 ' 7m n 4" <0 m7n¨ . (. m 0 )$ # '< *Q/,- 1 < ' > M= $ Y = >[ $ " $ * '<3H*_.% )&!8 *f1' !8 ) c "4 9) #')" $ i . (0 2 ) 9 1 . ) $U(' k ) Tl\n ) > * ') ^ = > M $ Y =3* [ $ b!$ $ H ') );& >') "c. 2 ')2!# )8ST!*_.¯. % $ pT - 9J$ +*) 0 0S'X. #; ( 0 W <' m m (' m7 0 * ) ^ *)= > M= $ Y = [ #f$ E') P ') #f$ c " $ ! (.) 1 4 1\«\m (' m7 0 2_ . ¨ 1 . _ . %(0 n31 9R0< ;o" 2m[2NY = 1 = 132 2 0 =?0 m7n (' m7 0 2L( 0 0 <=?' m 0 )$ E ').*_')o') '< ?]0 ^ > MM $ Y =)5 [ na$ c $ # pH $9R0 - 0S'X. (= n ' m7 0 ' ' k 1 ilk 13 . " `_ ! (.) 1 4 1\«\m (' m7 0 2 $ . 9 ^V@ ')9 ]0 h^ > M=)5$ Y = [ na$ c $ # pH $ ¨ . (M MNma;om ' " ilk 13 . " 2 (0 9R0< ;o" ' m7n K m71 $ ')*a2 9 x} "A3o,- ')*a2 9 x $ &$ > MM $ Y = [ na$ c $ # pH '< $ $ $ @') #'< ') $ $ = ; ' m m 1 0 2m7 0< ; 4m = 2m7 0 ¨ . \n313232\132 $ . 9 ^A@ ')9 P]} -^ > M M!$ Y = [ $ $ $ @'< #') ') $# 3H*_.% )}.& 2')2"" Y')QED 4#') '< ! $ 0 ) ¨ =?. 1 9 ; ) $&(' k ) > M ) * > ^ *)= > M $ ( Y == [ $ $ $ @') ') ') $ ! (.) 1J41\«\m (' m7 0 2 (0 9( ;om7n (' m7 0 2 $ & # , 1'< ". 1' > M)=)5$

Y M >[ Y M3* [ YM [

* 1') 9

! Q * > ) *}* ^ L* 5 * > M $ ! *Q 1') 9

* > ) 54MM ^ > * > M $ #f$na$ c 4 $ 9 "*_a ! *Q 1') 9

. &H & & & $ ilk ) ¨ . (M M ) 9( ; ) *<5 ) u^ 3M * > M M!$ #f$na$ c 4 $ 9 "*_a ! *Q 1') 9

. &H & @ $ ilk ) ¨ . (M M ) 9 ; ) * ) * > ^ * 5 > M3= *H$ #f$na$ c 4 $ !m m ' ilk 13 . 1 2 0 ! (.) 14 1\«\m (' m7 0 2S_ 4 # [') *Q?,-2" n > MM $

1'< !c8!) '< &% '<#" 1'< !c8!) '< &% '<#" 1'< !c8!) '< &% '<#" . $U(. /VL«T ' \n k 2 ' m7nY¨ . \n313232\132 $

M

Y M)5 [ $ $ O ')2 $ %R% 9 ! "%[')3R% . ') $ +*) # ?= 0 n ' ) 9R0< ; ) >}> & ) }* * ^ *<5 > MM3*H$ Y M [ a$gO ! > MM $ ,- '< %*Q*a"'< $

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