Probabilit y

http://dx.doi.org/10.1090/gsm/080

This page intentionally left blank

Probabilit y

Davar Khoshnevisan

Graduate Studies

in Mathematics

Volume 80

J^| American Mathematical Society Providence, Rhode Island

Editoria l B o a r d

David Co x Walter Crai g

Nikolai Ivano v Steven G . Krant z

David Saltma n (Chair )

2000 Mathematics Subject Classification. Primar y 60-01 ; Secondary 60-03 , 28-01 , 28-03 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /gsm-80

Library o f Congres s Cataloging-in-Publicat io n D a t a

Khoshnevisan, Davar . Probability / Dava r Khoshnevisan .

p. cm . — (Graduat e studie s i n mathematics , ISS N 1065-733 9 ; v. 80 ) Includes bibliographica l reference s an d index . ISBN-13: 978-0-8218-4215- 7 (alk . paper ) ISBN-10: 0-8218-4215- 3 (alk . paper ) 1. Probabilities . I . Title .

QA273.K488 200 7 519.2—dc22 200605260 3

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o reprint-permission@ams.org .

(c) 200 7 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t http : //www. ams. org/

10 9 8 7 6 5 4 3 2 1 1 2 1 1 1 0 0 9 0 8 0 7

To my famil y

This page intentionally left blank

Contents

Preface x i

General Notatio n x v

Chapter 1 . Classica l Probabilit y 1

§1. Discret e Probabilit y 1

§2. Conditiona l Probabilit y 4

§3. Independenc e 6

§4. Discret e Distribution s 6

§5. Absolutel y Continuou s Distribution s 1 0

§6. Expectatio n an d Varianc e 1 2

Problems 1 3

Notes 1 5

Chapter 2 . Bernoull i Trial s 1 7

§1. Th e Classica l Theorem s 1 8

Problems 2 1

Notes 2 2

Chapter 3 . Measur e Theor y 2 3

§1. Measur e Space s 2 3

§2. Lebesgu e Measur e 2 5

§3. Completio n 2 8

§4. Proo f o f Caratheodory' s Theore m 3 0

Problems 3 3

viii Contents

Notes 3 4

Chapter 4 . Integratio n 3 5

§1. Measurabl e Function s 3 5

§2. Th e Abstrac t Integra l 3 7

§3. 77-Space s 3 9

§4. Mode s o f Convergenc e 4 3

§5. Limi t Theorem s 4 5

§6. Th e Radon-Nikody m Theore m 4 7

Problems 4 9

Notes 5 2

Chapter 5 . Produc t Space s 5 3

§1. Finit e Product s 5 3

§2. Infinit e Product s 5 8

§3. Complement : Proo f o f Kolmogorov' s Extensio n Theore m . . . 6 0

Problems 6 2

Notes 6 4

Chapter 6 . Independenc e 6 5

§1. Rando m Variable s an d Distribution s 6 5

§2. Independen t Rando m Variable s 6 7

§3. A n Instructiv e Exampl e 7 1

§4. Khintchine' s Wea k La w o f Larg e Number s 7 1

§5. Kolmogorov' s Stron g La w o f Larg e Number s 7 3

§6. Application s 7 7

Problems 8 4

Notes 8 9

Chapter 7 . Th e Centra l Limi t Theore m 9 1

§1. Wea k Convergenc e 9 1

§2. Wea k Convergenc e an d Compact-Suppor t Function s 9 4

§3. Harmoni c Analysi s i n Dimensio n On e 9 6

§4. Th e Planchere l Theore m 9 7

§5. Th e 1- D Centra l Limi t Theore m 10 0

§6. Complement s t o th e CL T 10 1

Problems I l l

Notes 11 7

Contents i x

Chapter 8 . Martingale s 11 9

§1. Conditiona l Expectation s 11 9

§2. Filtration s an d Semi-Martingale s 12 6

§3. Stoppin g Time s an d Optiona l Stoppin g 12 9

§4. Application s t o Rando m Walk s 13 1

§5. Inequalitie s an d Convergenc e 13 4

§6. Furthe r Application s 13 6

Problems 15 1

Notes 15 7

Chapter 9 . Brownia n Motio n 15 9

§1. Gaussia n Processe s 16 0

§2. Wiener' s Construction : Brownia n Motio n o n [ 0 ,1) 16 5

§3. Nowhere-Differentiabilit y 16 8

§4. Th e Brownia n Filtratio n an d Stoppin g Time s 17 0

§5. Th e Stron g Marko v Propert y 17 3

§6. Th e Reflectio n Principl e 17 5

Problems 17 6

Notes 18 0

Chapter 10 . Terminus : Stochasti c Integratio n 18 1

§1. Th e Indefinit e It o Integra l 18 1

§2. Continuou s Martingale s i n L 2(P) 18 7

§3. Th e Definit e It o Integra l 18 9

§4. Quadrati c Variatio n 19 2

§5. Ito' s Formul a an d Tw o Application s 19 3

Problems 19 9

Notes 20 1

Appendix 20 3

§1. Hilber t Space s 20 3

§2. Fourie r Serie s 20 5

Bibliography 20 9

Index 21 7

This page intentionally left blank

Preface

Say what you know, do what you must, come what may.

-Sofya Kovalevskay a

To us probability is the very guide of life.

-Bishop Josep h Butle r

A few years ago the University of Utah switched fro m th e quarter syste m to the semester system. Thi s change gave the faculty a chance to re-evaluat e their cours e offerings . A s par t o f thi s re-evaluatio n proces s w e decide d t o replace th e usua l year-lon g graduat e cours e i n probabilit y theor y wit h on e that wa s a semeste r long . Ther e wa s goo d reaso n t o d o so . Th e rol e o f probability i n mathematics , science , an d engineerin g was , an d stil l is , o n the rise . Ther e i s increasin g deman d fo r a graduat e cours e i n probability . And ye t th e typica l graduat e studen t i s no t abl e t o tackl e a larg e numbe r of year-lon g course s outsid e hi s o r he r ow n researc h area . Thus , w e wer e presented wit h a non-trivial challenge : Ca n we offer a course that addresse s the need s o f ou r own , a s wel l a s other , graduat e students , al l withi n th e temporal confines of one semester? I believe that th e answer to the precedin g question i s "yes. "

This book presents a cohesive graduate course in measure-theoretic prob-ability tha t specificall y ha s th e one-semeste r studen t i n mind . Ther e is , i n fact, ampl e material to cover an ordinary year-long course at a more leisurely pace. See , fo r example , th e man y section s tha t ar e entitle d complements , and thos e tha t ar e calle d applications . However , th e primar y goal s o f thi s book ar e t o maintai n brevit y an d conciseness , an d t o introduc e probabilit y quickly an d a t a modestl y dee p level . I hav e use d a s m y mode l a stan -dard one-semeste r undergraduate cours e i n probability . I n tha t setting , th e

XI

Xll Preface

instructional issue s ar e wel l understood , an d mos t expert s agre e o n wha t should b e taught .

Giving a one-semeste r introductio n t o graduat e probabilit y necessaril y involves makin g concessions . Min e for m th e content s o f thi s book : N o men -tion i s mad e o f Kolmogorov' s theor y o f rando m series ; Levy' s continuit y theorem o f characteristi c function s i s sadl y omitted ; Marko v chain s ar e no t treated a t all ; an d th e constructio n o f Brownia n motio n i s Fourier-analyti c rather tha n "probabilistic. "

That i s no t t o sa y tha t ther e i s littl e coverag e o f th e theor y o f stochasti c processes. Fo r example , include d yo u wil l fin d a n introductio n t o Ito' s sto -chastic calculu s an d it s connection s t o ellipti c partia l differentia l equations . This topi c ma y see m ambitious , an d i t probabl y i s fo r som e readers . How -ever, m y experienc e i n teachin g thi s materia l ha s bee n tha t th e reade r wh o knows som e measur e theor y ca n cove r th e boo k u p t o an d includin g th e las t chapter i n a singl e semester . Thos e wh o wis h t o lear n measur e theor y fro m this boo k woul d probabl y ai m t o cove r les s stochasti c processes .

Teaching R e c o m m e n d a t i o n s . I n m y ow n lecture s I ofte n begi n wit h Chapter 2 and prov e th e D e Moivre-Laplac e centra l limi t theore m i n detail . Then, I spen d tw o o r thre e week s goin g ove r basi c result s i n analysi s [Chap -ters 3 throug h 5] . Onl y a handfu l o f th e sai d result s ar e actuall y proved . Without exception , on e o f the m i s Caratheodory' s monoton e clas s theore m (p. 30) . Th e fundamenta l notio n o f independenc e i s introduced , an d a num -ber o f importan t example s ar e worke d out . Amon g the m ar e th e wea k an d the stron g law s o f larg e number s [Chapte r 6] , respectivel y du e t o A . Ya . Khintchine an d A . N . Kolmogorov . Nex t follo w element s o f harmoni c anal -ysis an d th e centra l limi t theore m [Chapte r 7] . A majorit y o f the subsequen t lectures concer n J . L . Doob' s theor y o f martingale s (1940 ) an d it s variou s applications [Chapte r 8] . Afte r martingales , ther e ma y b e enoug h tim e lef t to introduc e Brownia n motio n [Chapte r 9] , construc t stochasti c integrals , and deduc e a strikin g computation , du e t o Chun g (1947) , o f th e distribu -tion o f th e exi t tim e fro m [—1,1 ] o f Brownia n motio n (p . 197) . I f a t al l possible, th e latte r topi c shoul d no t b e missed .

My persona l teachin g philosoph y i s t o showcas e th e bi g idea s o f proba -bility b y derivin g ver y few , bu t central , theorems . Frangoi s Mari e Aroue t [Voltaire] onc e wrot e tha t "th e ar t o f bein g a bor e i s t o tel l everything. " Viewed i n thi s light , a chie f ai m o f thi s boo k i s t o no t bore .

I woul d lik e t o leav e th e reade r wit h on e piec e o f advic e o n ho w t o bes t use thi s book . Rea d i t thoughtfully , an d wit h pe n an d paper .

Preface xm

Acknowledgements : Thi s boo k i s base d o n th e combine d content s o f sev -eral o f m y previou s graduat e course s i n probabilit y theory . Man y o f thes e were give n a t th e Universit y o f Uta h durin g th e pas t decad e o r so . Also , I have use d part s o f som e lecture s tha t I gav e durin g th e formativ e stage s o f my caree r a t MI T an d th e Universit y o f Washington . I wis h t o than k al l three institution s fo r thei r hospitalit y an d support , an d th e Nationa l Scienc e Foundation, th e Nationa l Securit y Agency , an d th e North-Atlanti c Treat y Organization fo r thei r financia l suppor t o f m y researc h ove r th e years .

All scholar s kno w abou t th e merit s o f librar y research . Nevertheless , the rol e o f thi s lor e i s underplaye d i n som e academi c texts . I , fo r one , foun d the followin g t o b e enlightening : Billingsle y (1995) , Breima n (1992) , Cho w and Teiche r (1997) , Chun g (1974) , Crame r (1936) , Dudle y (2002) , Durret t (1996), Fristed t an d Gra y (1997) , Gnedenk o (1967) , Karlin an d Taylo r (1975 , 1981), Kolmogoro v (1933 , 1950) , Krickeber g (1963 , 1965) , Lang e (2003) , Pollard (2002) , Resnic k (1999) , Strooc k (1993) , Varadha n (2001) , William s (1991), an d Woodroof e (1975) . Withou t doubt , ther e ar e othe r excellen t references. Th e studen t i s encourage d t o consul t othe r resource s i n additio n to th e presen t text . H e o r sh e woul d d o wel l t o remembe r tha t i t ma y b e nice t o kno w facts , bu t i t i s vitall y importan t t o hav e a perspective .

I a m gratefu l t o th e followin g fo r thei r variou s contribution s t o th e de -velopment o f thi s book : Nelso n Beebe , Rober t Brooks , Piete r Bowman , Re x Butler, Edwar d Dunne , Stewar t Ethier , Victo r Gabrenas , Pran k Gao , An a Meda Guardiola , Ja n Hannig , Henry k Hecht , Lajo s Horvath , Zsuzsann a Horvath, Ada m Keenan , Kari m Khader , Remigiju s Leipus , A n Le , Davi d Levin, Michae l Purcell , Pejma n Mahboubi , Pedr o Mendez , Ji m Pi tman , Na -talya Pluzhnikov , Matthe w Reimherr , Shang-Yua n Shiu , Jose f Steinebach , James Turner , Joh n Walsh , Ju n Zhang , an d Lian g Zhang . Man y o f thes e people hav e helpe d find typographica l errors , an d eve n a fe w seriou s mis -takes. Al l error s tha t remai n ar e o f cours e mine .

My famil y ha s bee n a stalwar t pilla r o f patience . Thei r kindnes s an d love wer e indispensabl e i n completin g thi s project . I than k the m deeply .

And las t bu t certainl y no t th e least , m y eterna l gratitud e i s extended t o my teachers , pas t an d present , fo r introducin g m e to the joys of mathematics . I hop e onl y tha t som e o f thei r ingenuit y an d spiri t persist s throughou t thes e pages.

Davar Khoshnevisa n Salt Lak e City , Januar y 200 7

This page intentionally left blank

General Notatio n

Here we se t fort h som e o f th e genera l notatio n tha t i s consistentl y use d i n the entir e book . Thi s i s standard mathematica l notation , an d w e may refe r to i t withou t furthe r mention .

Logic an d Se t Theory . Throughout , U and D respectively denot e unio n and intersection, and C the subset relation. Occasionall y we may write " a : = 6," wher e a and b could be sets , numbers , logica l expressions, functions , etc . Depending on the context, this may mean either "defin e a to be 6," or "defin e b to b e a. " W e will not mak e a distinctio n betwee n th e two .

If A,B C X , the n A c denote s th e complemen t o f A [i n X] . Th e de -pendence o n X i s usuall y suppresse d a s i t i s clea r fro m th e context . Le t A \ B : = A H 5C , an d A A B : = {A \ B) U (B \ A). Th e latte r i s called th e set difference o f A an d B.

A set i s denumerable i f i t i s either countabl e o r finite.

We frequently writ e "iff " a s short-hand fo r "i f an d onl y if. "

Finally, "V " and "3 " respectively stan d fo r "fo r all " an d "ther e exists. "

Euclidean Spaces . Throughout , R = (—o o , oo) denote s th e rea l line , Z = {0, ± 1, ±2 , . . .} th e integers , N = {1 ,2 , . . . } th e natura l numbers , an d Q denotes the rationals. I f X designate s any one of these, then X+ denote s th e non-negative element s o f X , an d X _ denote s th e non-positiv e elements . I f k G N the n X ^ denote s th e collectio n o f al l fc-tuples {x\ , . . . , Xk) suc h tha t # i , . . . , Xk are i n X. Fo r instance , R̂ _ denote s th e collectio n o f al l k- vectors that ar e non-negative coordinatewise . Th e comple x plane i s denoted b y C .

xv

XVI General Notation

If x , y G R , the n w e writ e x A y = min(x , y) fo r th e minimu m an d x V y = max(x , y) fo r th e maximum . Similarly , su p an d in f respectivel y refer t o supremu m an d infimu m operations .

Functions. I f X an d Y ar e two sets , then " / : X — > Y" stand s fo r " / map s X int o Y," an d "x >— • / (#) " refer s t o the ma p fro m x t o f(x). I f / : X — >• Y and A C Y , the n f~ 1{A) = {x G X : f(x) G ^4}. Thi s i s the inverse image of A.

The Big-O/Little- o Notation . Suppos e ai , «2 , . . ., &i, &2> • • • £ R » W e say that a n ~ 6 n [a s n — > oo ] when lim n_*oo(an/6n) = 1 . Whe n th e b^s ar e als o non-negative, "limsup n_^00 \a n\/bn < oo " i s often writte n a s "a n = 0(6 n)," and "lim^-^o o |an | /6 n = 0 " a s "a n = o(6 n)." Not e tha t a n = 0(b n) if f ther e exists a constan t C suc h tha t \a n\ < Cbn fo r al l n > 1 .

The big-O/little- o notatio n i s also applicable t o functions : u f(x) ~ #(x ) as x - ^ a " mean s a limx_^a(/(x)/^(x)) = 1" ; an d whe n g > 0 w e ma y write u f(x) — 0(g(x)) a s x — » a" fo r "limsup x_>a | / (x) | = 0(#(x)), " an d "/(#) = o(<7(#) ) a s x — > a" i n place o f "lim x-+a(f(x)/g(x)) — 0. "

Bibliography

Adams, W . J . (1974) . The Life and Times of the Central Limit Theorem. Ne w York : Kaedmo n Publishing Co .

Aldous, D . J . (1985) . Exchangeabili t y an d Relate d Topics . I n Ecole d'Ete de Probabilites de Saint-Flour, XIII —1983, Volum e 111 7 o f Lecture Notes in Math., pp . 1-198 . Berlin : Springer .

Alon, N . an d J . H . Spence r (1991) . The Probabilistic Method (Firs t ed.) . Ne w York : Wiley .

Andre, D . (1887) . Solutio n direct e d u problem e resol u pa r M . Ber t rand . C. R. Acad. Sci. Paris 105, 436-437.

Azuma, K . (1967) . Weighte d sum s o f certai n dependen t rando m variables . Tohoku Math. J. (2) 19, 357-367.

Bachelier, L . (1900) . Theori e d e l a speculation . Ann. Sci. Ecole Norm. Sup. 17, 21-86 . Se e als o th e 1995 reprint . Sceaux : Gauthier-Vi l lars .

Bachelier, L . (1964) . Theor y o f speculation . I n P . H . Cootne r (Ed.) , The Random Character of Stock Market Prices, pp . 17-78 . MI T Press . Translate d fro m Frenc h b y A . Jame s Boness .

Banach, S . (1931) . Ube r di e Baire'sch e kategori e gewisse r Funkionenmengen . Studia. Math. Ill, 174-179 .

Bass, R . F . (1998) . Diffusions and Elliptic Operators. Ne w York : Springer-Verlag .

Baxter, M . an d A . Renni e (1996) . Financial Calculus: An Introduction to Derivative Pricing. Cam -bridge: Cambridg e Universit y Press . Secon d (1998 ) reprint .

Berkes, I . (1998) . Result s an d problem s relate d t o th e pointwis e centra l limi t theorem . I n Asymptotic Methods in Probability and Statistics (Ottawa, ON, 1997), pp . 59—96 . Amsterdam : North-Holland .

Bernoulli, J . (1713) . Ars Conjectandi (The Art of Conjecture). Basel : Basileae : Impensi s Thurnis ioru m Fraetrum.

Bernstein, S . N . (1912/1913) . Demonstra t io n d u theorem e d e Weierstras s fonde e su r l e calcu l de s probabilites. Comm. Soc. Math. Kharkow 13, 1-2 .

Bernstein, S . N . (1964) . O n th e propert y characterist i c o f th e norma l law . I n Sobranie sochinenii. Tom IV: Teoriya veroyatnostei. Matematicheskaya Statistika. 1911-1946. "Nauka" , Moscow .

Billingsley, P . (1995) . Probability and Measure (Thir d ed.) . Ne w York : Joh n Wile y & Son s Inc .

Bingham, N . H . (1986) . Variant s o n th e la w o f th e i terate d logari thm . Bull. London Math. Soc. 18(5), 433-467.

Birkhoff, G . D . (1931) . Proo f o f th e ergodi c theorem . Proc. Nat. Acad. Sci. 17, 656-660 .

Black, F . an d M . Schole s (1973) . Pricin g o f option s an d corporat e liabilities . J. Political Econ. 81, 637-654.

Blumenthal , R . M . (1957) . A n extende d Marko v property . Trans. Amer. Math. Soc. 85, 52-72 .

Borel, E . (1909) . Le s probabilite s denombrable s e t leur s application s ar i thmetique . Rend. Circ. Mat. Palermo 27, 247-271 .

Borel, E . (1925) . Mecanique Statistique Classique (Thir d ed.) . Paris : Gauthier-Vi l lars .

209

210 Bibliography

Bourke, C. , J . M . Hitchcock, an d N . V. Vinodchandran (2005) . Entrop y rate s and finite-state dimension . Theoret. Comput. Sci. 349(3), 392-406 .

Bovier, A . an d P . Picc o (1996) . Limi t theorem s fo r Bernoull i convolutions . I n Disordered Systems (Temuco, 1991/1992), Volum e 53 , pp. 135-158 . Paris : Hermann .

Breiman, L . (1992) . Probability. Philadelphia , PA : Societ y fo r Industria l an d Applie d Mathematic s (SIAM). Se e also th e correcte d reprin t o f the origina l (1968) .

Bretagnolle, J . an d D . Dacunha-Castell e (1969) . Application s radonifiante s dan s le s espace s d e typ e p. C. R. Acad. Sci. Paris Ser. A-B 269, A1132-A1134 .

Broadbent, S . R . an d J . M . Hammersle y (1957) . Percolatio n processes . I . Crystal s an d mazes . Proc. Cambridge Philos. Soc. 53, 629-641 .

Buczolich, Z . and R . D . Mauldi n (1999) . O n th e convergenc e o f £)^L i f(nx) fo r measurabl e functions . Mathematika 46(2), 337-341 .

Burkholder, D . L . (1962) . Successiv e conditiona l expectation s o f a n integrabl e function . Ann. Math. Statist. 33, 887-893 .

Burkholder, D . L. , B . J . Davis , an d R . F . Gund y (1972) . Integra l inequalitie s fo r conve x function s o f operators o n martingales . I n Proc. Sixth Berkeley Symp. Math. Statist. Probab., Vol. II, Berkeley , Calif., pp . 223-240 . Univ . Californi a Press .

Burkholder, D . L . an d R . F . Gund y (1970) . Extrapolatio n an d interpolatio n o f quasi-linea r operator s on martingales . Acta Math. 124, 249-304 .

Cantelli, F . P . (1917a) . S u du e applicazion i d'u n teorem a d i G . Bool e ali a statistic a matematica . Atti della Reale Accademia Nationale dei Lincei, Serie V, Rendicotti 26, 295-302 .

Cantelli, F . P . (1917b) . Sull a probabilit a com e limit e dell a frequenze . Atti della Reale Accademia Nationale dei Lincei, Serie V, Rendicotti 26, 39-45 .

Cantelli, F . P . (1933a) . Considerazion i sull a legg e uniform e de i grand i numer i e sull a generalizzazion e di u n fondamental e teorem a de l signo r Levy . Giomale d. Istituto Italiano Attuari 4, 327-350 .

Cantelli, F . P . (1933b) . Sull a determinazion e empiric a dell e legg i d i probabilita . Giomale d. Istituto Italiano Attuari 4, 421-424 .

Caratheodory, C . (1948) . Vorlesungen iiber reelle Funktionen. Ne w York: Chelse a Publishing Company .

Champernowne, D . G . (1933) . Th e constructio n o f decimal s norma l i n th e scal e o f ten . J. London Math. Soc. 8, 254-260.

Chatterji, S . D . (1968) . Martingal e convergenc e an d th e Radon-Nikody m theore m i n Banac h spaces . Math. Scand. 22, 21-41 .

Chebyshev, P . L . (1846) . Demonstratio n elementair e d'un e propositio n general e d e l a theorie de s prob -ability. Crelle J. Math. 33(2), 259-267 .

Chebyshev, P . L . (1867) . De s valeurs moyennes . J. Math. Pures Appl. 12(2), 177-184 .

Chernoff, H . (1952) . A measur e o f asymptoti c efficienc y fo r test s o f a hypothesi s base d o n th e su m o f observations. Ann. Math. Statist. 23, 493-507 .

Chow, Y. S. and H . Teicher (1997) . Probability Theory: Independence, Inter changeability, Martingales (Third ed.) . Ne w York : Springer-Verlag .

Chung, K . L . (1947) . O n th e maximu m partia l su m o f independent rando m variables . Proc. Nat. Acad. Sci. U.S.A. 33, 132-136 .

Chung, K . L . (1974) . A Course in Probability Theory (Secon d ed.) . Ne w York-London : Academi c Press.

Chung, K.-L . an d P . Erdo s (1947) . O n th e lowe r limi t o f sum s o f independen t rando m variables . Ann. of Math. (2) 48, 1003-1013 .

Chung, K . L . an d P . Erdo s (1952) . O n th e applicatio n o f th e Borel-Cantell i lemma . Trans. Amer. Math. Soc. 72, 179-186 .

Ciesielski, Z . an d S . J . Taylo r (1962) . Firs t passag e time s an d sojour n time s fo r Brownia n motio n i n space an d th e exac t Hausdorf f measur e o f the sampl e path . Trans. Amer. Math. Soc. 103, 434-450 .

Cifarelli, D . M. and E . Regazzin i (1996) . D e Finetti' s contributio n t o probabilit y an d statistics . Statist. Sci. 11(4), 253-282 .

Coifman, R . R . (1972) . Distributio n functio n inequalitie s fo r singula r integrals . Proc. Nat. Acad. Sci. U.S.A. 69, 2838-2839 .

Copeland, A . H . an d P . Erdo s (1946) . Not e o n norma l numbers . Bull. Amer. Math. Soc. 52, 857-860 . Courtault, J.-M. , Y . Kabanov, B . Bru, P . Crepel, I . Lebon, an d A . Le Marchand (2000) . Loui s Bachelier .

On th e Centenar y o f Theorie de la Speculation. Math. Finance 10(3), 341-353 . Cover, T . M . an d J . A . Thoma s (1991) . Elements of Information Theory. Ne w York : Joh n Wile y Sz

Sons Inc .

Bibliography 211

Cox, J . C , S . A . Ross , an d M . Rubenstei n (1979) . Opt io n pricing : a simplifie d approach . J. Financial Econ. 7, 229-263 .

Cramer, H . (1936) . Ube r ein e Eigenschaf t de r normale n Verteilungsfunktion . Math. Z. J^l, 405-415 .

Csorgo, M . an d P . Reves z (1981) . Strong Approximations in Probability and Statistics. Ne w York : Academic Pres s Inc . [Harcour t Brac e Jovanovic h Publishers] .

de Acosta , A . (1983) . A ne w proo f o f th e Har tman-Win tne r la w o f th e i terate d logarithm . Ann. Probab. 11(2), 270-276 .

de Finet t i , B . (1937) . L a prevision : se s loi s logiques , se s source s subjectives . Ann. Inst. H. Poincare 7, 1-68.

de Moivre , A . (1718) . The Doctrine of Chances; or a Method of Calculating the Probabilities of Events in Play (Firs t ed.) . London : W . Pearson .

de Moivre , A . (1733) . Approximati o a d Summa m terminoru m Binomi i ( a + b) n i n Seriu m expansi . Privately Printed .

de Moivre , A . (1738) . The Doctrine of Chances; or a Method of Calculating the Probabilities of Events in Play (Secon d ed.) . London : H . Woodfall .

Dellacherie, C . an d P.-A . Meye r (1982) . Probabilities and Potential. B. Amsterdam : North-Hollan d Publishing Co . Theor y o f martingales , Translate d fro m Frenc h b y J . P . Wilson .

Devaney, R . L . (2003) . An Introduction to Chaotic Dynamical Systems. Boulder , CO : Westvie w Press . Reprint o f th e secon d (1989 ) edition .

Diaconis, P . an d D . Freedma n (1987) . A doze n d e Finett i-styl e result s i n searc h o f a theory . Ann. Inst. H. Poincare Probab. Statist. 23(2, suppl.) , 397-423 .

Diaconis, P . an d J . B . Kelle r (1989) . Fai r dice . Amer. Math. Monthly 96(4), 337-339 .

Donoho, D . L . an d P . B . Star k (1989) . Uncertaint y principle s an d signa l recovery . SI AM J. Appl. Math. 49(3), 906-931 .

Doob, J . L . (1940) . Regularit y propert ie s o f certai n familie s o f chanc e variables . Trans. Amer. Math. Soc. 47, 455-486 .

Doob, J . L . (1949) . Applicatio n o f th e theor y o f mart ingales . I n Le Calcul des Probabilites et ses Applications, pp . 23-27 . Paris : Centr e Nationa l d e l a Recherch e Scientifique .

Doob, J . L . (1953) . Stochastic Processes. Ne w York : Joh n Wile y 8z Sons Inc .

Doob, J . L . (1971) . Wha t i s a mart ingale ? Amer. Math. Monthly 78, 451-463 .

Dubins, L . E . an d D . A . Freedma n (1965) . A sharpe r for m o f th e Borel-Cantell i lemm a an d th e s tron g law. Ann. Math. Statist. 36, 800-807 .

Dubins, L . E . an d D . A . Freedma n (1966) . O n th e expecte d valu e o f a s toppe d mart ingale . Ann. Math. Statist 37, 1505-1509 .

Dudley, R . M . (1967) . O n predictio n theor y fo r nonsta t ionar y sequences . I n Proc. Fifth Berkeley Symp. Math. Statist. Probab., Vol. II, pp . 223-234 . Berkeley , Calif. : Univ . Californi a Press .

Dudley, R . M . (2002) . Real Analysis and Probability. Cambridge : Cambridg e Universit y Press . Revise d reprint o f th e 198 9 original .

Durret t , R . (1996) . Probability: Theory and Examples (Secon d ed.) . Belmont , CA : Duxbur y Press .

Dynkin, E . an d A . Jushkevic h (1956) . Stron g Marko v processes . Teor. Veroyatnost. i Primenen. 1, 149-155.

Erdos, P . (1948) . Som e remark s o n th e theor y o f graphs . Bull. Amer. Math. Soc. 53, 292-294 .

Erdos, P . (1949) . O n th e stron g la w o f larg e numbers . Trans. Amer. Math. Soc. 67, 51—56 .

Erdos, P . an d G . Szekere s (1935) . A combinatoria l proble m i n geometry . Composito. Math. 2, 463—470 .

Erdos, P . an d A . Reny i (1959) . O n Cantor ' s serie s wit h convergen t ^ l/q-n- Ann. Univ. Sci. Budapest. Eotvos. Sect. Math. 2, 93-109 .

Erdos, P . an d A . Reny i (1970) . O n a ne w la w o f larg e numbers . J. Analyse Math. 23, 103-111 .

Etemadi , N . (1981) . A n elementar y proo f o f th e s t ron g la w o f larg e numbers . Z. Wahrsch. Verw. Geb. 55, 119-122 .

Falconer, K . J . (1986) . The Geometry of Fractal Sets, Volum e 85 . Cambridge : Cambridg e Universit y Press.

Fatou, P . J . L . (1906) . Serie s tr igonometrique s e t serie s d e Taylor . Acta Math. 69, 372-433 .

Feller, W . (1945) . Th e fundamenta l limi t theorem s i n probability . Bulletin A.M.S. 51, 800-832 .

Feller, W . (1955a) . O n differentia l operator s an d boundar y conditions . Comm. Pure Appl. Math. 8, 203-216.

Feller, W . (1955b) . O n secon d orde r differentia l operators . Ann. of Math. (2) 61, 90-105 .

212 Bibliography

Feller, W . (1956) . O n generalize d Sturm-Liouvill e operators . I n Proceedings of the Conference on Differential Equations (dedicated to A. Weinstein), pp . 251—270 . Universit y o f Marylan d Boo k Store, Colleg e Park , Md .

Feller, W . (1957) . An Introduction to Probability Theory and Its Applications. Vol. I (Secon d ed.) . New York : Joh n Wile y & Son s Inc .

Feller, W . (1966) . An Introduction to Probability Theory and Its Applications. Vol. II. Ne w York : John Wile y & Son s Inc .

Fortuin, C . M. , P . W . Kasteleyn , an d J . Ginibr e (1971) . Correlatio n inequalitie s o n som e partiall y ordered sets . Comm. Math. Phys. 22, 536-564 .

Frechet, M . R . (1930) . Su r l a convergenc e e n probabili te . Metron 8, 1-48 .

Freiling, C . (1986) . Axiom s o f symmetry : Throwin g dar t s a t th e rea l numbe r line . J. Symbolic Logic 5 / ( 1 ) , 190-200 .

Fristedt, B . an d L . Gra y (1997) . A Modern Approach to Probability Theory. Boston , MA : Birkhause r Boston Inc .

Garsia, A . M . (1965) . A simpl e proo f o f E . Hopf' s maxima l ergodi c theorem . J. Math. Mech. 14, 381 382 .

Georgii, H.-O . (1988) . Gibbs Measures and Phase Transitions. Berlin : Walte r d e Gruyte r & Co .

Gerber, H . U . an d S.-Y . R . L i (1981) . Th e occurrenc e o f sequenc e pa t te rn s i n repeate d experiment s an d hi t t ing t ime s i n a Marko v chain . Stochastic Process. Appl. 11(1), 101-108 .

Glivenko, V . (1933) . Sull a determinazion e empiric a dell e legg i d i probabili ty . Giornale d. Istituto Italiano Attuari 4, 92-99 .

Glivenko, V . (1936) . Su l teorem a limit e dell a teori a dell e funzion i carat teris t iche . Giornale d. Instituto Italiano attuari 7, 160-167 .

Gnedenko, B . V . (1967) . The Theory of Probability. Ne w York : Chelse a Publishin g Co . Translate d from th e fourt h Russia n editio n b y B . D . Seckler .

Gnedenko, B . V . (1969) . O n Hilbert ' s Sixt h Proble m (Russian) . I n Hubert's Problems (Russian) , pp . 116-120. "Nauka" , Moscow .

Gnedenko, B . V . an d A . N . Kolmogoro v (1968) . Limit Distributions for Sums of Independent Random Variables. Reading , Mass. : Addison-Wesle y Publishin g Co . Transla te d fro m th e origina l Russian , annota ted , an d revise d b y Ka i La i Chung . Wi t h appendice s b y J . L . Doo b an d P . L . Hsu . Revise d edition.

Gr immet t , G . (1999) . Percolation (Secon d ed.) . Berlin : Springer-Verlag .

Hamedani , G . G . an d G . G . Walte r (1984) . A fixe d poin t theore m an d it s applicatio n t o th e centra l limit theorem . Arch. Math. (Basel) 43(3), 258-264 .

Hammersley, J . M . (1963) . A Mont e Carl o solutio n o f percolatio n i n a cubi c lat t ice . I n S . F . B . Alde r and M . Rotenber g (Eds.) , Methods in Computational Physics, Volum e I . London : Academi c Press .

Hardy, G . H . an d J . E . Littlewoo d (1914) . Som e problem s o f diophantin e approximation . Acta Math. 31, 155-239.

Hardy, G . H . an d J . E . Littlewoo d (1930) . A maxima l theore m wit h function-theoreti c applications . Acta Math. 54, 81-166 .

Harris, T . E . (1960) . A lowe r boun d fo r th e critica l probabil i t y i n a certai n percolatio n process . Proc. Cambridge Philos. Soc. 56, 13-20 .

Harrison, J . M . an d D . Krep s (1979) . Martingale s an d arbi t rag e i n mult iperio d securitie s markets . J. Econ. Theory 20, 381-408 .

Harrison, J . M . an d S . R . Plisk a (1981) . Martingale s an d stochasti c integral s i n th e theor y o f continuou s t rading. Stoch. Proc. Their Appl. 11, 215-260 .

Har tman, P . an d A . Wintne r (1941) . O n th e la w o f th e i terate d logarithm . Amer. J. Math. 63, 169-176 .

Hausdorff, F . (1927) . Mengenlehre. Berlin : Walte r D e Gruyte r & Co .

Hausdorff, F . (1949) . Grundziige der Mengenlehre. Ne w York : Chelse a publishin g Co .

Helms, L . L . an d P . A . Loe b (1982) . A nonstandar d proo f o f th e mart ingal e convergenc e theorem . Rocky Mountain J. Math. 12(1), 165-170 .

Hoeffding, W . (1963) . Probabil i t y inequalitie s fo r sum s o f bounde d rando m variables . Amer. Statist. Assoc. 58, 13-30 .

Hoeffding, W . (1971) . Th e L\ nor m o f th e approximatio n erro r fo r Bernstein- typ e polynomials . J . Approximation Theory 4i 347-356 .

Houdre, C , V . Perez-Abreu , an d D . Surgaili s (1998) . Interpolat ion , correlatio n identities , an d inequal -ities fo r infinitel y divisibl e variables . J. Fourier Anal. Appl. ^ ( 6 ) , 651-668 .

Hunt, G . A . (1956) . Som e theorem s concernin g Brownia n motion . Trans. Amer. Math. Soc. 81, 294-319 .

Hunt, G . A . (1957) . Markof f processe s an d potentials . I , II . Illinois J. Math. 1, 44-93 , 316-369 .

Bibliography 213

Hunt, G . A . (1966) . Martingales et processus de Markov. Paris : Dunod .

Ionescu Tulcea , A . an d C . Ionesc u Tulce a (1963) . Abstrac t ergodi c theorems . Trans. Amer. Math. Soc. 107, 107-124 .

Isaac, R . (1965) . A proo f o f th e mart ingal e convergenc e theorem . Proc. Amer. Math. Soc. 16, 842-844 .

I to, K . (1944) . Stochasti c integral . Proc. Imp. Acad. Tokyo 20, 519-524 .

Jones, R . L . (1997/1998) . Ergodi c theor y an d connection s wit h analysi s an d probability . New York J. Math. 3A (Proceeding s o f th e Ne w Yor k Journa l o f Mathemat ic s Conference , Jun e 9-13 , 1997) , 31-6 7 (electronic).

Kac, M . (1937) . Un e remarqu e su r le s polynome s d e M.S . Bernstein . Studia Math. 7, 49 -51 .

Kac, M . (1939) . O n a characterizatio n o f th e norma l distr ibution . Amer. J. Math. 61, 726-728 .

Kac, M . (1949) . O n deviation s betwee n theoretica l an d empirica l distr ibutions . Proc. Nat. Acad. Sci. U.S.A. 35, 252-257 .

Kac, M . (1956) . Foundation s o f kineti c thoery . I n J . Neyma n (Ed.) , Proc. Third Berkeley Symp. on Math. Statist. Probab., Volum e 3 , pp . 171-197 . Univ . o f Calif .

Kahane, J .-P . (1997) . A fe w generi c propert ie s o f Fourie r an d Taylo r series . I n Trends in Probability and Related Analysis (Taipei, 1996), pp . 187-196 . Rive r Edge , NJ : Worl d Sci . Publishing .

Kahane, J .-P . (2000) . Baire' s categor y theore m an d tr igonometri c series . J. Anal. Math. 80, 143-182 .

Kahane, J .-P . (2001) . Probabili t ie s an d Baire ' s theor y i n harmoni c analysis . I n Twentieth Century Harmonic Analysis —A Celebration (II Ciocco, 2000), Volum e 3 3 o f NATO Sci. Ser. II Math. Phys. Chem., pp . 57-72 . Dordrecht : Kluwe r Acad . Publ .

Karlin, S . an d H . M . Taylo r (1975) . A First Course in Stochastic Processes (Secon d ed.) . Academi c Press [Harcour t Brac e Jovanovic h Publishers] , Ne w York-London .

Karlin, S . an d H . M . Taylo r (1981) . A Second Course in Stochastic Processes. Ne w York : Academi c Press Inc . [Harcour t Brac e Jovanovic h Publishers] .

Keller, J . B . (1986) . Th e probabili t y o f heads . Amer. Math. Monthly 93(3), 191-197 .

Kesten, H . (1980) . Th e critica l probabil i t y o f bon d percolatio n o n th e squar e latt ic e equal s ^ . Comm. Math. Phys. 7^(1) , 41-59 .

Khintchine, A . an d A . Kolmogoro v (1925) . Ube r Konvergen z vo n Reihe n dere n Gliede r durc h de n Zufal l bes t immt werden . Rec. Math. Moscow 32, 668-677 .

Khintchine, A . Y . (1923) . Ube r dyadisch e Bruche . Math. Z. 18, 109-116 .

Khintchine, A . Y . (1924) . Ei n Sat z de r Wahrscheinlichkeitsrechnung . Fund. Math. 6, 9-10 .

Khintchine, A . Y . (1929) . Su r l a lo i de s grand s nombres . C. R. Acad. Sci. Paris 188, 477-479 .

Khintchine, A . Y . (1933) . Asymptotische Gesetz der Wahrscheinlichkeitsrechnung. Springer .

Kinney, J . R . (1953) . Continuit y propert ie s o f sampl e function s o f Marko v processes . Trans. Amer. Math. Soc. 74, 280-302 .

Knight, F . B . (1962) . O n th e rando m wal k an d Brownia n motion . Trans. Amer. Math. Soc. 103, 218-228.

Knight, F . B . (1981) . Essentials of Brownian Motion and Diffusion. Providence , R.I. : America n Mathematical Society .

Knuth , D . E . (1981) . The Art of Computer Programming. Vol. 2 (Secon d ed.) . Reading , Mass. : Addison-Wesley Publishin g Co . Seminumerica l Algorithms .

Kochen, S . an d C . J . Ston e (1964) . A not e o n th e Borel-Cantel l i lemma . Illinois J. Math. 8, 248-251 .

Kolmogorov, A . (1930) . Su r l a lo i fort e de s grande s nombres . C. R. Acad. Sci. Paris 191, 910-911 .

Kolmogorov, A . N . (1929) . Ube r da s Geset z de s iterierte n Logari thmus . Math. Ann. 101, 126-135 .

Kolmogorov, A . N . (1933) . Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin : Springer .

Kolmogorov, A . N . (1950) . Foundations of Probability. Ne w York : Chelse a Publishi g Company . Trans -lation edite d b y Natha n Morrison .

Krickeberg, K . (1963) . Wahrscheinlichkeitstheorie. S tu t tga r t : Teubner .

Krickeberg, K . (1965) . Probability Theory. Reading , Massachusetts : Addison-Wesley .

Kyburg, Jr. , H . E . an d H . E . Smokie r (1980) . Studies in Subjective Probability (Secon d ed.) . Hunt ington , N.Y.: Rober t E . Kriege r Publishin g Co . Inc .

Lacey, M . T . an d W . Phil ip p (1990) . A not e o n th e almos t sur e centra l limi t theorem . Statist. Probab. Lett. 9(3), 201-205 .

Lamb, C . W . (1973) . A shor t proo f o f th e mart ingal e convergenc e theorem . Proc. Amer. Math. Soc. 38, 215-217.

Lange, K . (2003) . Applied Probability. Ne w York : Springer-Verlag .

214 Bibliography

Laplace, P . S . (1782) . Memoir e su r le s approximation s de s formule s qu i son t fonction s d e t res-grand s nombres. Technica l report , Histoir e d e I 'Academi e Royal e de s Science s d e Par is .

Laplace, P.-S . (1805) . Traite de Mecanique Celeste, Volum e 4 . Che z J . B . M . Dupra t , a n 7 (Crapelet ) . Reprinted b y th e Chelse a Publishin g Co . (1967) . Translate d b y N . Bowditch .

Laplace, P.-S . (1812) . Theorie Analytique des Probabilites, Vol. I and II. V[ieme ] Courcier . Reprinte d in Oeuvres completes de Laplace, Volum e VI I (1886) , Paris : Gauthier-Vil lars .

Lebesgue, H . (1910) . Su r l ' integratio n de s fonction s discontinues . Ann. Ecole Norm. Sup. 27(3), 361-450.

Levi, B . (1906) . Sopr a l ' integrazion e dell e serie . Rend. Instituto Lombardino di Sci. e Lett. 39(2), 775-780.

Levy, P . (1925) . Calcul des Probabilites. Paris : Gauthier-Vi l lars .

Levy, P . (1937) . Theorie de I Addition des Variables Aleatoires. Paris : Gauthier -Vi l lars .

Levy, P . (1951) . L a mesur e d e Hausdorf f d e l a courb e d u mouvemen t brownie n a n dimensions . C. R. Acad. Sci. Paris 233, 600-602 .

Li, S.-Y . R . (1980) . A mart ingal e approac h t o th e s tud y o f occurrenc e o f sequenc e pa t t e rn s i n repeate d experiments. Ann. Probab. 8(6), 1171-1176 .

Liapounov, A . M . (1900) . Su r un e proposit io n d e l a theori e de s probabil i tes . Bulletin de I'Academie Imperiale des Sciences de St. Petersbourg 13(A), 359-386 .

Liapounov, A . M . (1922) . Nouvell e form e d u theorem e su r l a limit e d e probability . Memoires de I'Academie Imperiale des Sciences de St. Petersbourg 12(5), 1-24 .

Lindeberg, J . W . (1922) . Ein e neu e Herleitun g de s Exponentialgesetze s i n de r Wahrscheinlichkeitsrech -nung. Math. Z. 15, 211-225 .

Lindvall, T . (1982) . Bernstei n polynomial s an d th e la w o f larg e numbers . Math. Sci. 7(2), 127-139 .

Lipschitz, R . (1876) . Su r l a possibilit e d' integre r complemen t u n system e donn e d 'equat ion s differentielles. Bull. Sci. Math. 10, 149-159 .

Mahmoud, H . M . (2000) . Sorting: A distribution theory. Wiley-Interscience , Ne w York .

Markov, A . A . (1910) . Recherche s su r u n ca s remarquabl e d'epreuve s dependantes . Acta Math. 33, 87-104.

Mat tner , L . (1999) . Produc t measurability , paramete r integrals , an d a Fubini-Tonel l i counterexample .

Enseign. Math. (2) ^5(3-4) , 271-279 .

Mazurkiewicz, S . (1931) . Su r le s fonction s no n derivables . Studia. Math. Ill, 92-94 . McShane, E . J . (1969) . A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and sto-

chastic integrals. Memoir s o f the America n Mathemat ica l Society , No . 88 . Providence , R.I. : America n Mathemat ica l Society .

Merton, R . C . (1973) . Theor y o f rationa l optio n pricing . Bell J. of Econ. and Management Sci. ^ (1 ) , 141-183.

Mukherjea, A . (1972) . A remar k o n Tonelli ' s theore m o n integratio n i n produc t spaces . Pacific J. Math. 42, 177-185 .

Mukherjea, A . (1973/1974) . Remar k o n Tonelli ' s theore m o n integratio n i n produc t spaces . II . Indiana Univ. Math. J. 23, 679-684 .

Nash, J . (1958) . Continuit y o f solution s o f paraboli c an d ellipti c equations . Amer. J. Math. 80, 931-954 .

Nelson, E . (1967) . Dynamical Theories of Brownian Motion. Pr inceton , N.J . : Pr inceto n Universit y Press.

Norris, J . R . (1998) . Markov Chains. Cambridge : Cambridg e Universit y Press . Reprin t o f 199 7 original .

Okamoto, M . (1958) . Som e inequalitie s relatin g t o th e part ia l su m o f binomia l probabili t ies . Ann. Inst. Statist. Math. 10, 29-35 .

Paley, R . E . A . C , N . Wiener , an d A . Zygmun d (1933) . Note s o n rando m functions . Math. Z. 37, 647-668.

Paley, R . E . A . C . an d A . Zygmun d (1932) . A not e o n analyti c function s i n th e uni t circle . Proc. Camb. Phil. Soc. 28, 366-372 .

Perrin, J . B . (1913) . Les Atomes. Paris : Librairi e F . Alcan . Se e als o Atoms. Th e revise d reprin t o f th e second (1923 ) Englis h edition . Va n Nostrand , Ne w York . Translate d b y Dalzie l Llewelly n Hammick .

P i tman , J . (1981) . A not e o n L-2 maximal inequalities . I n Seminar on Probability, XV (Univ. Stras-bourg, Strasbourg, 1979/1980) (French), Volum e 85 0 o f Lecture Notes in Math., pp . 251-258 . Berlin : Springer.

Plancherel, M . (1910) . Contr ibut io n a I 'etud e d e l a representat io n d 'un e fonctio n arbi t rair e pa r de s integrates definies . Rend. Circ. Mat. Palermo 30, 289-335 .

Plancherel, M . (1933) . Su r le s formule s d e reciprocit y d u typ e d e Fourier . J. London Math. Soc. 8, 220-226.

Bibliography 215

Plancherel, M . an d G . Poly a (1931) . Su r le s valeur s moyenne s de s fonction s reelle s definie s pou r toute s les valuer s d e l a variables . Comment. Math. Helv. 3, 114-121 .

Poincare, H . (1912) . Calcul des Probabilites. Paris : Gauthier-Villars .

Pollard, D . (2002) . A User's Guide to Measure Theoretic Probability. Cambridge : Cambridg e Univer -sity Press .

Polya, G . (1920) . Ube r de n zentrale n Grenzwertsat z de r Wahrscheinlichkeitstheori e un d da s Momenten -problem. Math. Zeit. 8, 171-181 .

Rademacher, H . (1919) . Ube r partiell e un d total e Differenzierbarkei t I . Math. Ann. 89, 340-359 .

Rademacher, H . (1922) . Einig e Satz e ube r Reihe n vo n allgemeine n Orthogonalfunktionen . Math. Ann. 87, 112-138 .

Raikov, D . (1936) . O n som e ari thmetica l propert ie s o f summabl e functions . Math. Sb. 1(43:3), 377-383 .

Ramsey, F . P . (1930) . O n a proble m o f forma l logic . Proc. London Math. Soc. 30(2), 264-286 .

Renyi, A . (1962) . Wahrscheinlichkeitsrechnung. Mit einem Anhang uber Informationstheorie. Berlin : VEB Deutsche r Verla g de r Wissenschaften .

Resnick, S . I . (1999) . A Probability Path. Boston , MA : Birkhause r Bosto n Inc .

Revuz, D . an d M . Yo r (1999) . Continuous Martingales and Brownian Motion (Thir d ed.) . Berlin : Springer-Verlag.

Rosenlicht, M . (1972) . Integratio n i n finit e te rms . Amer. Math. Monthly 79, 963-972 .

Schnorr, C.-P . an d H . Stim m (1971/1972) . Endlich e Automate n un d Zufallsfolgen . Acta Informat. 1(4), 345-359.

Schrodinger, E . (1946) . Statistical Thermodynamics. Cambridge : Cambridg e Universit y Press .

Shannon, C . E . (1948) . A mathemat ica l theor y o f communication . Bell System Tech. J. 27, 379-423 , 623-656.

Shannon, C . E . an d W . Weave r (1949) . The Mathematical Theory of Communication. Urbana , 111. : Univ. o f Illinoi s Press .

Shultz, H . S . an d B . Leonar d (1989) . Unexpecte d occurence s o f th e numbe r e . Math. Magazine 62(4), 269-271.

Sierpinski, W . (1920) . Su r le s rappor t entr e l 'existenc e de s integrale s f Q f(x,y)dx, f Q f(x,y)dy e t

f* dxf£ f(x,y)dy. Fund. Math. 1, 142-147 .

Skolem, T . (1933) . Ei n kombinatorische r Sat z mi t anwendun g au f ei n logische s Entscheidungsproblem . Fund. Math. 20, 254-261 .

Skorohod, A . V . (1961) . Issledovaniya po teorii sluchainykh protsessov. Kiev . Univ. , Kiev . [Studies in the Theory of Random Processes. Translate d fro m Russia n b y Script a Technica , Inc. , Addison -Wesley Publishin g Co. , Inc. , Reading , Mass . (1965) . Se e als o th e secon d editio n (1985) , Dover , Ne w York.].

Skorohod, A . V . (1965) . Studies in the theory of random processes. Addison-Wesle y Publishin g Co. , Inc., Reading , Mass .

Slutsky, E . (1925) . Ube r stochastisch e Asymptote n un d Grenzwerte . Metron 5, 3-89 .

Solovay, R . M . (1970) . A mode l o f se t theor y i n whic h ever y se t o f real s i s Lebesgu e measurable . Ann. Math. 92, 1-56 .

Steinhaus, H . (1922) . Le s probabili te s denombrable s e t leu r rappor t a l a theori e d e mesure . Fund. Math. 4, 286-310 .

Steinhaus, H . (1930) . Su r l a probabili t e d e l a convergenc e d e serie . Studia Math. 2, 21-39 .

Stigler, S . M . (1986) . The History of Statistics. Cambridge , MA : Th e Belkna p Pres s o f Harvar d University Press .

Stirling, J . (1730) . Methodus Differentialis. London : Whis to n & : Whi te .

Strassen, V . (1967) . Almos t sur e behavio r o f sum s o f independen t rando m variable s an d mart ingales . I n Proc. Fifth Berkeley Sympos. Math. Statist, and Probability (Berkeley, Calif., 1965/66), pp . Vol . II, Pa r t 1 , pp . 315-343 . Berkeley , Calif. : Univ . Californi a Press .

Stroock, D . W . (1993) . Probability Theory, An Analytic View. Cambridge : Cambridg e Universit y Press.

Stroock, D . W . an d O . Zeitoun i (1991) . Microcanonica l dis tr ibut ions , Gibb s s ta tes , an d th e equivalenc e of ensembles . I n Random Walks, Brownian Motion, and Interacting Particle Systems, pp . 399—424 . Boston, MA : Birkhause r Boston .

Tandori , K . (1983) . Th e Lif e an d Work s o f Lipo t Fejer . I n Functions, series, operators, Vol. I, II (Budapest, 1980), Volum e 3 5 o f Colloq. Math. Soc. Jdnos Bolyai, pp . 77—85 . Amsterdam : North -Holland.

Trotter , H . F . (1959) . A n elementar y proo f o f th e centra l limi t theorem . Arch. Math. 10, 226-234 .

216 Bibliography

Turing, A . M . (1934) . O n th e Gaussia n erro r function . Technica l report , Unpublishe d Fellowshi p Dissertation, King' s Colleg e Library , Cambridge .

Varadhan, S . R . S . (2001) . Probability Theory. Ne w York : Ne w Yor k Universit y Couran t Ins t i tu t e o f Mathemat ica l Sciences .

Varberg, D . E . (1966) . Convergenc e o f quadrat i c form s i n independen t rando m variables . Ann. Math. Statist. 37, 567-576 .

Veech, W . A . (1967) . A Second Course in Complex Analysis. Ne w York , Amsterdam : W . A . Benjamin , Inc.

Ville, J . (1939) . Etude Critique de la Notion de Collectif. Paris : Gauthier-Vi l lars .

Ville, J . (1943) . Su r 1'application , a u n criter e d ' independance , d u denombremen t de s inversion s presentees pa r un e permuta t ion . C. R. Acad. Sci. Paris 217, 41-42 .

von Neumann , J . (1940) . O n ring s o f operators , III . Ann. Math. 41, 94-161 .

von Smoluchowski , M . (1918) . Ube r de n Begrif f de s Zufall s un d de n Ursprun g de r Wahrscheinlichkeit . Die Naturwissenschaften 6(17), 253-263 .

Wagon, S . (1985) . I s n normal ? Math. Intelligencer 7(3) , 65-67 .

Wiener, N . (1923a) . Differentia l space . J. Math. Phys. 2, 131-174 .

Wiener, N . (1923b) . Th e homogeneou s chaos . Amer. J. Math. 60, 879-036 .

Williams, D . (1991) . Probability with Martingales. Cambridge : Cambridg e Universit y Press .

Wong, C . S . (1977) . Classroo m Notes : A Not e o n th e Centra l Limi t Theorem . Amer. Math. Monthly 84(6), 472 .

Woodroofe, M . (1975) . Probability with Applications. Ne w York : McGraw-Hil l Boo k Co .

Young, L . C . (1970) . Som e ne w stochasti c integral s an d Stieltje s integrals . I . Analogue s o f Hardy -Littlewood classes . I n Advances in Probability and Related Topics, Vol. 2, pp . 161—240 . Ne w York : Dekker.

Zabell, S . L . (1995) . Ala n Tur in g an d th e centra l limi t theorem . Amer. Math. Monthly 102(6), 483-494 .

Index

Symbols &(Q), Bore l sigma-algebr a o n Q 2 4 Bin(n , p), binomia l distributio n 8 C(X) , continuou s function s fro m X t o R 4 9 C, th e comple x number s x v Cb(X), bounde d continuou s function s fro m

X t o R 9 4 CcpQ, compactl y supporte d continuou s func -

tions fro m X t o R 9 4 C£°(R f c), infinitel y differentiat e function s

of compac t suppor t fro m R fc t o R 11 1 Cov(X , V), covarianc e betwee n X an d Y 6 7 EX, expectatio n o f X 3 9 E[X; A] = E[X1 A] = f AXdP 3 9 Geom(p), geometri c distributio n 8 L p , rando m variable s wit h p finit e absolut e

moments 3 9 i fP, completio n o f LP 4 3 N(fj,, cr 2), norma l distributio n 1 1 N , th e natura l number s x v ^ ( n ) , th e powe r se t o f Cl 2 4 Poiss(A), Poisso n distributio n 9 Q, th e rational s x v R, th e rea l number s x v S n _ 1 , th e uni t spher e i n R n 10 2 SD(X), standar d deviatio n o f X 6 7 Unif(a, 6) , unifor m distributio n 1 1 VarX, varianc e o f X 6 7 X + , th e positiv e element s o f X x v Z, th e integer s x v a.e., almos t everywher e 4 3 a.s., almos t surel y 4 3 dv/dfj,, Radon—Nikody m derivativ e 4 7 / + = max( / ,0 ) 3 8 / - = m a x ( - / ,0 ) 3 8

i.i.d., independen t identicall y distribute d 6 8 m c ( ^ ) , monoton e clas s generate d b y srf 3 0

imiLP(M) = (/ i / i p<w1 / p 3 9 ||X||p = (EIIXIP]) 1/** 3 9 i / < ^ , absolut e continuit y 4 7 <r(X), c r ( ^ ) , . .. sigma-algebr a generate d b y

X , ^ , et c 24 , 49 , 12 0 A, V, th e mi n an d ma x operator s xv i

A Absolute continuit y . . 11 , see also Measur e Adams, Willia m J 2 2 Adapted proces s 126 , 18 2 Aldous, Davi d J 15 7 Almost everywher e convergenc e 4 3 Almost sur e

central limi t theore m 9 0 convergence 4 3

Alon, Nog a 8 9 Andre, Desir e 179 , 18 0 Approximation t o th e identit y 20 5 Avogadro, Lorenz o Roman o Amede o Carl o

159 Azuma, Kazuok i 15 7 Azuma-Hoeffding inequalit y . . 155, see also

Heoffding's inequalit y

B Bachelier, Loui s Jean Baptist e Alphons e 159 ,

177, 18 0 Backward [o r reversed ] martingal e 15 5 Banach, Stefa n 15 7 Bass, Richar d Frankli n 20 1 Baxter, Marti n 15 7 Berkes, Istva n 9 0

217

218 Index

Bernoulli trial s 8 Bernoulli, Jaco b 18 , 22 , 7 1 Bernstein

polynomials 7 7 proof o f th e Weierstras s theore m 7 7

Bernstein [Bernshtein] , Sergei Natanovich 77 , 117

Bessel's inequalit y 20 8 Bessel, Priedric h Wilhel m 20 8 Bingham, Nichola s H 15 7 Binomial distributio n 8 , see also

Distribution Birkhoff, Georg e Davi d 9 0 Black, Fische r 144 , 14 5 Black-Scholes Formul a 14 5 Blumenthal's zero-on e la w 17 7 Blumenthal, Rober t M 177 , 18 0 Borel se t and/o r sigma-algebr a 2 4 Borel, Emil e . . 52 , 73 , 86, 89 , 109 , 117 , 13 7 Borel-Cantelli lemm a 63 , 73 , see also

Paley-Zygmund inequalit y Dubins-Freedman 15 6 Levy's 13 6

Borel-Caratheodory lemm a 10 9 Bounded convergenc e theore m 4 5

conditional 12 1 Bourke, Chri s 9 0 Bovier Anto n 11 3 Bretagnolle, Jea n 15 8 Broadbent, S . R 8 3 Brown, Rober t 15 9 Brownian bridg e 17 7 Brownian motio n

and th e hea t equatio n 177 , 178 , 196 , 19 8 as a Gaussia n proces s 16 3 Einstein's predicate s 16 0 exit distributio n . . 196, 197 , 200 , see also

Chung's formul a filtration 171 , see also Filtratio n gambler's rui n formul a 20 0 nowhere differentiabilit y o f 16 8 quadratic variatio n 16 3 Wiener's constructio n 166-16 8 with drif t 20 1

Bru, Bernar d 18 0 Buczolich, Zolta n 9 0 Bunyakovsky, Vikto r Yakovlevic h 4 0 Burkholder, Donal d L 52 , 15 6

Call option s 14 4 Cantelli, Francesc o Paol o 73 , 80, 85 , 89 , 13 8 Cantor se t 8 8 Cantor, Geor g 3 4 Cantor—Lebesgue functio n 8 8 Caratheodory Extensio n Theore m 2 7

Caratheodory, Constantin e 27 , 10 9 Cauchy

sequence 4 6 summability tes t 18 3

Cauchy, Augusti n Loui s . .40 , 108 , 113 , 18 3 Cauchy-Bunyakovsky-Schwarz inequalit y 4 0 Central limi t theore m . 19, 22 , 89 , 100 , 10 2

Bovier-Picco 11 3 de Moivre-Laplac e 1 9 Liapounov's 11 4 Lindeberg's 11 5 projective 10 2 via Liapounov' s metho d 10 5 Ville's . . . 1 1 6 with erro r estimate s 105 , 11 6

Champernowne, Davi d Gawe n 9 0 Characteristic functio n 96 , 96-11 7

convergence theore m 99 , 10 2 inversion theore m 11 2 uniqueness theore m 9 9

Chatterji, Srisht i Dha v 15 7 Chebyshev's inequalit y 18 , 4 3

conditional 15 2 for sum s 6 3

Chebyshev, Pafnuti i Lvovic h . . . . 18 , 43, 6 3 Chernoff's inequalit y 5 1 Chernoff, Herma n 51 , 52, 8 7 Chung's formul a 197 , 20 0 Chung, Ka i La i xii , 89 , 19 7 Ciesielski, Zbignie w 20 1 Cifarelli, Donat o Michel e 15 8 Coifman, Ronal d R 5 2 Compact suppor t [o r compactl y supported ]

function 9 4 Compact-support

process 18 2 Complete

measure spac e 2 8 topological spac e 4 2

Completion 2 9 Conditional expectatio n 120-12 5

and predictio n 12 2 classical 12 4 properties 120-121 , 12 3 towering propert y 12 3

Conditional probabilit y 4 , 12 5 Consistent measure s 5 9 Convergence

almost everywher e 4 3 almost sur e 4 3 in LP 4 3 in measur e 4 3 in probabilit y 4 3 weak 9 1

Convergence theore m 102 , see also Characteristic functio n

Convex functio n 40 , 50 , 15 6

Index 219

Convolution 15 , 98 , 11 3 Cootner, Pau l H 15 9 Copeland, Arthu r H 9 0 Correlation 6 7 Countable (sub- ) additivit y 2 4 Courtault, Jean-Miche l 18 0 Covariance 6 7

matrix 16 1 Cover, Thoma s M 8 9 Cox, Joh n C 14 4 Crepel, Pierr e 18 0 Cramer's theore m 10 7 Cramer, Haral d 102 , 107 , 11 7 Cramer-Wold devic e 10 2 Csorgo, Miklo s 15 7 Cumulative distributio n functio n . . . 66 , see

also Distributio n functio n Cylinder se t 5 8

D Dacunha-Castelle, Didie r 15 8 Davis, Burges s J 5 2 de Acosta , Alejandr o 156 , 15 8 de Finetti ' s theore m 15 6 de Finetti , Brun o 156-15 8 de Moivre , Abraha m 15 , 19 , 2 2 de Moivre-Laplac e centra l limi t theore m 19 ,

see also Centra l limi t theore m de Moivre' s formul a . 19, see also Stirling' s

formula Degenerate norma l 11 , see also Distributio n Dellacherie, Claud e 20 1 Density functio n 10 , 14 , 4 8 Devaney, Rober t L 9 0 Diaconis, Pers i 22 , 11 7 Dimension 5 9 Dini, Uliss e 18 2 Dini-continuous proces s

i n L 2 ( P ) 18 2 Distribution 49 , 6 5

binomial 8 characteristic functio n 9 7 connection t o Poisso n 10 , 1 9 mean an d varianc e 1 2

Cauchy 14 , 11 3 non-existence o f th e mea n 1 4

discrete 7 discrete unifor m 1 5 exponential 1 3

characteristic functio n 9 7 connection t o unifor m 1 5 mean an d varianc e 1 3

function 13 , 33, 6 6 gamma 1 3

characteristic functio n 11 2 mean an d varianc e 1 3

geometric 8

mean an d varianc e 1 2 hypergeometric 1 3

mean an d varianc e 1 3 infinitely divisibl e 11 3 negative binomia l 1 3

mean an d varianc e 1 3 normal 11 , 28

characteristic functio n 97 , 16 1 degenerate 1 1 mean an d varianc e 1 2 multi-dimensional 28 , 16 1 standard 1 1

Poisson 9 , 1 0 characteristic functio n 9 7 connection t o binomia l 10 , 1 9 mean an d varianc e 1 3

uniform 11 , 15 , 2 8 characteristic functio n 9 7 connection t o discret e unifor m 1 5 connection t o exponentia l 1 5 mean an d varianc e 1 2

Dominated convergenc e theore m 4 6 conditional 12 1

Donoho, Davi d L 11 5 Doob's

decomposition 128 , 15 2 martingale convergenc e theore m . . . . 13 4 martingales 127 , 20 0 maximal inequalit y 13 4

continuous-time 18 9 optional stoppin g theore m 13 0 strong (p , p)-inequality

continuous-time 18 9 strong .^-inequalit y 15 6 strong L p-inequality 15 3

Pitman's improvemen t 15 3 Doob, Josep h Le o . . xii , 127 , 130 , 134 , 152 ,

153, 156 , 157 , 189 , 20 0 Dubins, Leste r E 154 , 15 8 Dudley, Richar d Mansfiel d 22 , 5 1 Durrett, Richar d 90 , 15 8 Dyadic filtration 142 , 14 8 Dyadic interva l 142 , 14 8 Dynkin, Eugen e B 18 0

E Einstein, Alber t 15 9 Elementary functio n 3 7 Entropy 7 9 Erdos, Pau l [Pal ] 81 , 88-9 0 Etemadi, Nasrolla h 8 9 European option s 14 4 Event 35 , see also Measurabl e se t Exchangeable 155 , 15 6 Expectation 12 , 3 9

220 Index

F Falconer, Kennet h K 3 4 Fatou's lemm a 45 , 5 0

conditional 12 1 Fatou, Pierr e Josep h Loui s 45 , 5 2 Fejer, Leopol d 11 7 Feller, Willia m K 16 , 117 , 157 , 20 1 Fermi, Enric o 8 4 Filtration 12 6

Brownian 17 1 right-continuous 17 1

Fischer, Erns t Sigismun d 16 8 FKG inequalit y 6 3 Fortuin, C M 6 4 Fourier serie s 205-20 8 Fourier transfor m 96 , see also

Characteristic functio n Fourier, Jea n Baptist e Josep h 9 6 Frechet, Mauric e Ren e 5 2 Freedman, Davi d A 117 , 154 , 15 8 Freiling, Chri s 2 3 Fubini, Guid o 5 5 Fubini-Tonelli theore m 5 5

inapplicability o f 56-58 , 62 , 6 3

G Gambler's rui n formul a 133 , see

also Rando m walk , see also Brownia n motion

Garsia, Adrian o M 9 0 Gaussian proces s 16 3 Geometric distributio n 8 , see also

Distribution Georgii, Hans-Ott o 15 8 Gerber, Han s U 15 7 Ginibre, Jea n 6 4 Glivenko, Valeri i Ivanovic h 80 , 117 Glivenko-Cantelli theore m 8 0 Gnedenko, Bori s Vladimirovic h . . . . 52 , 11 7 Grimmett, Geoffre y R 8 3 Gundy, Richar d F 5 2

H Hadamard's Inequalit y 5 1 Hadamard, Jacque s Salomo n 5 1 Hamedani, Gholamhossei n Gharago z . . 117 Hammersley, Joh n M 83 , 8 9 Hardy, Godfre y Harol d . 138 , 140 , 143 , 18 7 Hardy-Littlewood maxima l functio n . . 143,

187 Harris, Theodor e E 8 3 Harrison, J . Michae l 14 4 Hartman, Phili p 13 8 Hausdorff measur e 3 4 Hausdorff, Feli x 34 , 13 8 Helms, Leste r L 157 Hilbert spac e 20 3

Hilbert, Davi d 5 2 Hitchcock, Joh n M 9 0 Hoeffding's inequalit y 51 , 87, see also

Azuma-Hoeffding Hoeffding, Wassil y 51 , 52 , 87 , 89 , 15 7 Holder

continuous functio n 78 , 19 9 inequality 3 9

conditional 12 1 generalized 5 1

Holder, Ott o Ludwi g 39 , 51 , 19 9 Houdre, Christia n 11 7 Hunt, Gilber t A 130 , 171 , 18 0

Independence 14 , 62 , 6 8 Indicator functio n 3 6 Infinitely divisibl e 11 3 Information inequalit y 8 7 Inner produc t 20 3 Integrable functio n 3 9 Integral 3 9 Inverse imag e xv i Inversion theore m 112 , see also

Characteristic functio n Ionescu Tulcea , Alexandr a 15 7 Ionescu Tulcea , Cassiu s 15 7 Isaac, Richar d 15 7 Ito

formula 194 , 19 5 integral 181-18 5

indefinite 18 5 under Dini-continuit y 18 4

isometry 18 4 lemma see also It o formul a

Ito, Kiyos i 181 , 184 , 185 , 194 , 19 5

Jensen's inequalit y 4 0 conditional 12 1

Jensen, Johan n Ludwi g Wilhel m Waldema r 40

Jones, Roge r L 5 2 Jushkevich [Yushkevich] , Alexande r A . 18 0

K Kabanov, Yur i 18 0 Kac, Mar k 15 , 78 , 115-11 7 Kahane, Jean-Pierr e 15 7 Kasteleyn, Piete r Wille m 6 4 Keller, Josep h B 2 2 Kesten, Harr y 8 3 Khintchine [Khinchin] , Aleksand r Yakovle -

vich xii , 71 , 72, 88 , 89 , 138, , 153, 177

Khintchine's inequality 88

Index 221

weak la w o f larg e number s . . 72 , see also Law o f larg e number s

Kinney, Joh n R 18 0 Knight, Fran k B 178 , 20 1 Knuth, Donal d E 8 4 Kochen, Simo n 8 9 Kolmogorov

consistency theore m 60 , see also Kolmogorov extensio n theore m

extension theore m 6 0 maximal inequalit y 7 4 one-series theore m 8 5 strong la w o f larg e number s . 73, see also

Law o f larg e number s zero-one la w 69 , 13 6

Kolmogorov, Andre i Nikolaevic h xii , 52 , 60 , 69, 73 , 74 , 85 , 89 , 103 , 117 , 138 , 15 3

Kreps, Davi d M 14 4 Krickeberg's decompositio n 12 8 Krickeberg, Klau s 12 8 Kyburg, Henr y E. , J r 15 8

Lacey, Michae l T 9 0 Lamb, Charle s W 15 7 Laplace, Pierre-Simo n . 16, 19 , 21 , 157 , 20 6 Law o f larg e number s 71—8 8

and Mont e Carl o simulatio n 8 3 and Shannon' s theore m 7 9 and th e Glivenko-Cantell i theore m . . . 8 0 Erdos-Renyi 8 8 strong 73 , 8 5 weak 7 2

Law o f rar e event s 1 9 Law o f th e iterate d logarith m 138 , 154 , 15 6

for Brownia n motio n 17 7 Law o f tota l probabilit y 5 Lebesgue

differentiation theore m 140 , 15 5 measurable 30 , see also Measurabl e measure 25 , see also Measur e

Lebesgue, Henr i Leo n 52 , 113 , 14 0 Lebon, Isabell e 18 0 Leonard, Bil l 8 6 Levi's monoton e convergenc e theore m . . 4 6 Levi, Bepp o 5 2 Levy's

Borel-Cantelli lemm a 13 6 concentration inequalit y 11 2 equivalence theore m 15 5 forgery theore m 17 8

Levy, Pau l . . 90 , 92 , 99 , 112 , 117 , 136 , 155 , 160, 178 , 196 , 20 0

Li, Rober t Shuo-Ye n 149 , 15 7 Liapounov [Lyapunov] , Aleksandr Mikhailovic h

104, 11 4 Lindeberg, Jar l Waldema r 104 , 11 5

Lindvall, Torgn y 8 9 Liouville, Josep h 11 , 16 , 10 8 Lipschitz continuou s 147 , 18 3 Lipschitz, Rudol f Ott o Sigismun d 15 7 Littlewood, Joh n Edenso r . . . 138 , 140 , 143 ,

187 Loeb, Pete r A 15 7

M Mahmoud, Hosa m M 15 7 Le Marchand , Arnau d 18 0 Markov

inequality 4 3 property . . . 174 , see also Stron g Marko v

of Brownia n motio n 160 , 16 3 Markov, Andre i Andreyevic h 43 , 8 9 Martingale 12 6

and likelihoo d ratio s 15 2 continuous-time 18 7 convergence theore m 134 , see also Doob' s representations 14 5 reversed 15 5 transforms 12 7

Mass functio n 7 , 1 4 Mattner, Lut z 6 4 Mauldin, R . Danie l 9 0 Maximal inequalit y . . . . 74-76 , 87 , 134 , 18 9 Maxwell, Jame s Cler k 11 7 Mazurkiewicz, Stefa n 15 7 McShane, Edwar d Jame s 19 9 Mean 39 , see also Expectatio n Measurable

function 3 5 Lebesgue 3 0 set 2 4 space 25 , see also Measur e spac e

Measure 2 4 absolutely continuou s 4 7 counting 3 3 finite produc t 5 4 Lebesgue 2 5

invariance propertie s 3 3 probability 2 5 space 2 5 support o f 33 , see also Suppor t

Merton, Rober t C 14 4 Meyer, Paul-Andr e 20 1 Minkowski's inequalit y 4 0

conditional 12 1 Minkowski, Herman n 4 0 Mixture 5 1 Modulus o f continuit y 77 , 18 2 Monotone clas s 3 0

theorem 3 0 Monotone convergenc e theore m 46 , see also

Levi conditional 12 1

222 Index

Monte Carl o integration 8 4 simulation 83 , see also La w o f larg e

numbers, 8 4 Mukherjea, Arunav a 6 4

N Nash's Poincar e inequalit y 11 6 Nash, Joh n 116 , 11 7 Negative par t o f a functio n 3 8 Nelson, Edwar d 18 0 Newton's metho d 8 7 Newton, Isaa c 8 7 Nikodym, Otto n Marci n 4 7 Normal distributio n 11 , see also

Distribution Normal number s 86 , 9 0 Norris, Jame s R 9 0

o Okamoto, Masash i 8 7 Optional stoppin g

continuous-time 187 Optional stoppin g theore m 13 0 Options 14 4 Orthogonal 20 3 Orthogonal decompositio n theore m . . . . 20 3

p Paley, Raymon d Edwar d Ala n Christophe r

86, 157 , 16 8 Paley-Zygmund inequalit y 8 6 Parseval de s Chenes , Marc-Antoin e . . . . 11 7 Parseval's identit y 11 7 Percolation 8 2 Perez-Abreu, Victo r 11 7 Perrin, Jea n Baptist e 18 0 Philipp, Walte r 9 0 Picco, Pierr e : 11 3 Piecewise continuou s functio n 1 1 Pitman's L 2 inequalit y 153 , see also

Doob's stron g L p inequalit y Pitman, Jame s W 15 3 Plancherel theore m 99 , 11 5 Plancherel, Miche l 34 , 97 , 11 5 Pliska, Stanle y R 14 4 Poincare inequalit y 11 6 Poincare, Jule s Henr i 16 , 22 , 15 9 Point-mass 2 5 Poisson distributio n 9 , see also Distributio n Poisson, Simeo n Deni s 9 , 1 9 Poissonization 1 0 Polya's urn s 15 2 Polya, Georg e 34 , 117 , 15 2 Positive par t o f a functio n 3 8 Positive-type functio n 11 3 Power se t 2 4

Previsible proces s 12 7 Probability spac e 25 , see also Measur e

space Product

measure 54 , see also Measur e topology 5 8

Q Quadratic variatio n 163 , see also Brownia n

motion

R Rademacher's theore m 14 7 Rademacher, Han s 89 , 14 7 Radon, Johan n 4 7 Radon-Nikodym theore m 4 7 Raikov's ergodi c theore m 88 , 9 0 Raikov, Dmitri i Abramovic h 88 , 9 0 Ramsey numbe r 8 1 Ramsey, Pran k Plumpto n 8 1 Random

permutation . . 3 , 9, 22 , 116 , 155 , see also Central limi t theorem , Ville' s

set 6 4 variable 3 5

absolutely continuou s 10 , 1 4 discrete 7 , 1 4

vector 1 4 Random wal k 13 1

gambler's rui n formul a 133 , 15 2 nearest neighborhoo d 13 2 simple 13 2

Reflection principl e 175 , 17 9 Regazzini, Eugeni o 15 8 Rennie, Andre w 15 7 Renyi, Alfre d 88 , 8 9 Reversed martingal e 15 5 Revesz, Pa l 15 7 Revuz, Danie l 20 1 Riemann, Geor g Friedric h Bernhar d . . . 11 3 Riemann-Lebesgue lemm a 11 3 Riesz reresentatio n theore m 4 9 Riesz, Prigye s 49 , 16 8 Rosenlicht, Maxwel l 1 6 Ross, Stephe n A 14 4 Rubenstein, Mar k 14 4

s Schnorr, Claus-Pete r 9 0 Scholes, Myro n 144 , 14 5 Schrodinger, Erwi n 8 6 Schwarz's lemm a 10 8 Schwarz, Herman n Amandu s 40 , 10 8 Semimartingale 12 6 Shannon's theore m 7 9 Shannon, Claud e Elwoo d 78 , 7 9 Shultz, Harri s S 8 6

Index 223

Sierpiriski, Wacla w 57 , 6 4 Sigma-algebra 2 3

Borel 2 4 generated b y a rando m variabl e .49 , 12 0

Simple functio n 3 7 Simple wal k . . . 114, see also Rando m walk ,

132 Simpson, Thoma s 2 2 Simulation 83 , see also La w o f larg e

numbers Skolem, Thoral f Alber t 8 1 Skorohod [Skorokhod] , Anatoli Vladimirovic h

116, 17 8 Skorohod embeddin g 17 8 Skorohod's theore m 11 6 Slutsky, Evgen y 50 , 5 2 Slutsky's theore m 5 0 Smokier, Howar d E 15 8 Solovay, Rober t M 23 , 3 4 Spencer, Joe l 8 9 Stable distributio n 17 8 Standard deviatio n 12 , 6 7 Standard norma l . 11 , see also Distributio n Stark, Phili p B 11 5 Steinhaus probabilit y spac e 4 4 Steinhaus, Hug o 44 , 89 , 138 , 15 4 Stigler, Steve n Mac k 8 9 Stimm, H 9 0 Stirling's formul a . .21, 22, 82 , 156 , see also

de Moivre' s formul a Stirling, Jame s 2 1 Stochastic integra l 179 , see also ltd integra l Stochastic proces s 12 6

continuous-time 18 1 Stone, Charle s J 8 9 Stopping tim e 129 , 17 0

simple 17 0 Strassen, Volke r 15 7 Strong la w o f larg e number s . . . 73 , see also

Law o f larg e number s Cantelli's 8 5 for dependen t variable s 8 5 for exchangeabl e variable s 15 5

Strong Marko v propert y 17 4 Stroock, Danie l W 11 7 Submartingale 12 6

continuous-time 20 0 Supermartingale 12 6

continuous-time 20 0 Support 3 3 Surgailis, Donata s 11 7 Szekeres, Gabo r [Gyorgy ] 8 1

Tandori, Karol y 11 7 Taylor, Samue l Jame s 20 1 Thomas, Jo y A 8 9

Tonelli, Leonid a 5 5 towering property o f conditional expectation s

123 Triangle inequalit y 38 , 4 1 Trigonometric polynomia l 20 5 Trotter, Hal e Freema n 11 7 Turing, Ala n Mathiso n 11 7 Turner, Jame s A 15 2

17 [/-statistics 151 , 15 5 Ulam, Stanisia w 8 4 Uncertainty principl e 51 , 52, 11 5 Uncorrelated rando m variable s 6 7 Uniform distributio n 11 , see also

Distribution on S " " 1 10 3

Uniform integrabilit y 5 1 , 15 4 and wea k convergenc e 11 5

Uniqueness theore m 99 , see also Characteristic functio n

V Varberg, Dal e E 15 1 Variance 12 , 6 7

conditional 15 2 Veech, Willia m A 11 7 Viete, Frangoi s 11 7 Ville, Jea n 117 , 15 7 Vinodchandran, N . V 9 0 von Neumann , Joh n 48 , 8 4 von Smoluchowski , Maria n 15 9

w Wagon, Stanle y 9 0 Wald's identit y 131 , 15 3 Wald, Abraha m 131 , 15 3 Wallis, Joh n 11 4 Walter, Gilber t G 11 7 Weak convergenc e 9 1 Weak la w o f larg e number s 72 , see also

Law o f larg e number s Weaver, Warre n 7 8 Weierstrass approximatio n theore m 77 , see

also Bernstei n Hoeffding's refinemen t 8 9 Kac's refinemen t 7 8

Weierstrass, Kar l Theodo r Wilhel m 7 7 Weyl, Herman n 5 2 White nois e 178 , 17 9 Wiener proces s . . . . 180 , see also Brownia n

motion Wiener, Norber t . . . 157 , 159 , 166 , 168 , 17 9 Williams, Davi d 15 7 Wintner, Aure l 13 8 Wold, Herma n O . A 10 2 Wong, Ch i Son g 2 2

224 Index

Woodroofe, Michae l 15 4

Y Yor, Mar c 20 1 Young integra l 199 , 20 0 Young's inequalit y 5 1 Young, Laurenc e Chishol m 19 9 Young, Willia m Henr y 5 1

z Zabell, Sand y L 11 7 Zeitouni, Ofe r 11 7 Zero-one la w

Blumenthal's 17 7 Kolmogorov's 69 , 13 6

Zygmund, Anton i 86 , 157 , 16 8