Course in Probability Theory

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3 5 8 1 CONDITIONING . MARKOV PROPERTY . MARTINGALESimilarly, generalize the downcrossing inequality for a positive supermartingale{X„ , as follows :X 1 Abv[Q b ] I h } ing of Theorems 9 .4 .2 and 9 .4 .3 we have, for a positivesupermartingale {X,,, n E N} :X A av ( Q,b} > k} < (b ) ( b )(n)J~{v{a b} > k}
9 .4 INEQUALITIES AND CONVERGENCE 1 359; 9 . Let IX,, : h'7-,,n E N} be a supermartingale satisfying the conditionlimn -0C (` (X„) > -oc . Then we have the representation X„ = X,, + X,7 where{X,,, c „} is a martingale and {X„, %,} is a positive supennartingale such thatlim„, 0 X„ = 0 in L 1 as well as a.e. This is the analogue of F . Riesz'sdecomposition of a superharmonic function, X ;, being the harmonic part andX;, the potential part . [HINT : Use Doob's decomposition X„ = Y„ - Z„ andput X„=Y„-~`(Z" I~'/n) •]10 . Let IX,,, ?X„ } be a potential ; namely a positive supermartingale suchthat Jim, ,--+,, = 0; and let X„ = Y„ - Z„ be the Doob decomposition[cf. (6) of Sec . 9 .3]. Show thatXn =I(Z,, I~T-n ) - Zn .* 11 . If {X„} is a martingale or positive submartingale such that(Xn) < oc, then {X„ } converges in L2 as well as a .e .SUN_12 . Let {~,,, n E N} be a sequence of independent and identicallydistributed r.v.'s with zero mean and unit variance ; and S„ _ ~~=.I ~jThen for any optional r .v . a relative to {~,, } such that c (ja- ) < oc, wehave c (IS a I) < /2-C`( .,Aa-) and c(S a ) = 0 . This is an extension of Wald'sequation due to Louis Gordon . [HINT : Truncate a and put 77k = (S2 /,/) -(Sn_ 1 /,/k - 1) ; thenk-1 fit k)77k dJ'00k=1k}/-1k- < 2(r( ,,fc-e ) ;now use Schwarz's inequality followed by Fatou's lemma .]The next two problems are meant to give an idea of the passage fromdiscrete parameter martingale theory to the continuous parameter theory .13 . Let {X t , `At ; t E [0, 1 ]} be a continuous parameter supermartingale .For each t E [0, 1 ] and sequence It,j decreasing tot, {X t „ } converges a .e . andin L 1 . For each t E [0, 1 ] and sequence {t,, } increasing to t, {X t, } converges a .e .but not necessarily in L 1 . [HINT : In the second case consider X t„ - (1'7{X t I ?It,, } .]*14. In Exercise 13 let Q be the set of rational numbers in [0, 1] . Foreach t E (0, 1) both limits below exist a .e . :lilnXs , limX s .'Tt,cQ[HINT : Let {Q,,, n > 1 } be finite subsets of Q such that Q,, T Q ; and apply theupcrossing inequality to {X,•, S E Q,, }, then let n ---> oo .],i tsEQ
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ACOURSE INPROBABILITYTHEORYTHIRD ED
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This book is printed on acid-free p
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vi I CONTENTS4 .3 Vague convergence
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Preface to the third editionIn this
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Preface to the second editionThis e
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Preface to the first editionA mathe
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PREFACE TO THE FIRST EDITION I xv(d
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Distribution function1 .1 Monotone
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1 .1 MONOTONE FUNCTIONS 1 3Example
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1 .1 MONOTONE FUNCTIONS 1 5havef I
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8 1 DISTRIBUTION FUNCTIONb ithe siz
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10 1 DISTRIBUTION FUNCTIONF2 - F ;
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12 ( DISTRIBUTION FUNCTIONThe next
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14 1 DISTRIBUTION FUNCTIONbecomes o
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Measure theory2 .1 Classes of setsL
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2 .1 CLASSES OF SETS 1 19and so = 6
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2.2 PROBABILITY MEASURES AND THEIR
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2 .2 PROBABILITY MEASURES AND THEIR
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2.2 PROBABILITY MEASURES AND THEIR
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3 .1 GENERAL DEFINITIONS 1 3 5of a
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3 .1 GENERAL DEFINITIONS 1 37Specif
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3 .1 GENERAL DEFINITIONS 1 39by (2)
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3 .2 PROPERTIES OF MATHEMATICAL EXP
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3.2 PROPERTIES OF MATHEMATICAL EXPE
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3 .3 INDEPENDENCE 1 533.3 Independe
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3 .3 INDEPENDENCE 1 55PROOF . We gi
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3 .3 INDEPENDENCE 1 57Corollary . I
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3 .3 INDEPENDENCE 1 59Then we haven
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3 .3 INDEPENDENCE 1 61in which ther
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3 .3 INDEPENDENCE 1 63We have thus
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3 .3 INDEPENDENCE I 65EXERCISES1 .
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3 .3 INDEPENDENCE I 67If do : {E~n
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4 .1 VARIOUS MODES OF CONVERGENCE 1
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4 .1 VARIOUS MODES OF CONVERGENCE I
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4 .1 VARIOUS MODES OF CONVERGENCE 1
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4 .2 ALMOST SURE CONVERGENCE; BOREL
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4.2 ALMOST SURE CONVERGENCE; BOREL-
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4 .2 ALMOST SURE CONVERGENCE ; BORE
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4 .3 VAGUE CONVERGENCE 1 85In this
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4 .3 VAGUE CONVERGENCE 1 87PROOF .
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4 .3 VAGUE CONVERGENCE ( 89By Sec .
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4.4 CONTINUATION 1 918. If g„ and
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4 .4 CONTINUATION 1 93that is linea
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4.4 CONTINUATION 1 95A family of p
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4 .4 CONTINUATION 1 97if X„ -+ X
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4 .5 UNIFORM INTEGRABILITY ; CONVER
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4.5 UNIFORM INTEGRABILITY ; CONVERG
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5 .1 SIMPLE LIMIT THEOREMS 1 107eve
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5 .1 SIMPLE LIMIT THEOREMS 1 109and
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5 .1 SIMPLE LIMIT THEOREMS 1 1 1 1r
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5.2 WEAK LAW OF LARGENUMBERS 1 1 1
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5.2 WEAK LAW OF LARGENUMBERS 1 1 1
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5.2 WEAK LAW OF LARGENUMBERS I 1 1
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5 .2 WEAK LAW OF LARGENUMBERS I 1 1
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5 .3 CONVERGENCE OF SERIES 1 12112.
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5 .3 CONVERGENCE OF SERIES 1 123whe
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5 .3 CONVERGENCE OF SERIES 1 1 2 5W
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5 .3 CONVERGENCE OF SERIES 1 127Thi
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5 .4 STRONG LAW OF LARGENUMBERS 1 1
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5.4 STRONG LAW OF LARGENUMBERS 1 13
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5.4 STRONG LAW OF LARGENUMBERS 1 13
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is an r cv) as follows co) (o) is c
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5 .5 APPLICATIONS 1 1 4 1PROOF . Su
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5 .5 APPLICATIONS 1 1 43null set Z
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5 .5 APPLICATIONS 1 1 45PROOF.Since
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5 .5 APPLICATIONS 1 1 4 7One should
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5 .5 APPLICATIONS 1 149Smile Borel,
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6.1 GENERAL PROPERTIES; CONVOLUTION
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6.1 GENERAL PROPERTIES ; CONVOLUTIO
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6 .1 GENERAL PROPERTIES ; CONVOLUTI
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CS6 .1 GENERAL PROPERTIES; CONVOLUT
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6 .1 GENERAL PROPERTIES ; CONVOLUTI
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x[ J6 .2 UNIQUENESS AND INVERSION 1
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6 .2 UNIQUENESS AND INVERSION 1 163
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6 .2 UNIQUENESS AND INVERSION 1 165
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I6 .2 UNIQUENESS AND INVERSION 1 1
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6 .3 CONVERGENCE THEOREMS I 169For
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6 .3 CONVERGENCE THEOREMS 1 171ther
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6.3 CONVERGENCE THEOREMS 1 173metri
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6 .4 SIMPLE APPLICATIONS I 175b > a
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6 .4 SIMPLE APPLICATIONS I 177k-1 j
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6 .4 SIMPLE APPLICATIONS 1 179Theor
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6 .4 SIMPLE APPLICATIONS 1 181then
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6 .4 SIMPLE APPLICATIONS 1 183We en
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6 .4 SIMPLE APPLICATIONS 1 185limit
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6 .5 REPRESENTATIVE THEOREMS 1 187c
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l6 .5 REPRESENTATIVE THEOREMS 1 189
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6 .5 REPRESENTATIVE THEOREMS 1 191(
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6 .5 REPRESENTATIVE THEOREMS I 193w
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6 .5 REPRESENTATIVE THEOREMS I 195E
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6 .6 MULTIDIMENSIONAL CASE; LAPLACE
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6 .6 MULTIDIMENSIONAL CASE ; LAPLAC
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6 .6 MULTIDIMENSIONAL CASE ; LAPLAC
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6 .6 MULTIDIMENSIONAL CASE; LAPLACE
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7Central limit theorem andits ramif
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7.1 LIAPOUNOV'S THEOREM l 207are "n
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7.1 LIAPOUNOV'S THEOREM 1 209the sa
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(D7.1 LIAPOUNOV'S THEOREM 1 211If1,
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7 .1 LIAPOUNOV'S THEOREM I 213and p
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7 .2 LINDEBERG-FELLER THEOREM 1 215
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7.2 LINDEBERG-FELLER THEOREM 1 217W
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(D if Sn /s'n does7.3 RAMIFICATIONS
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7.3 RAMIFICATIONS OF THE CENTRAL LI
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7 .3 RAMIFICATIONS OF THE CENTRAL L
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7 .4 ERROR ESTIMATION 1 2 35as n -~
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22x2x27 .4 ERROR ESTIMATION 1 23 7b
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7 .4 ERROR ESTIMATION l 23 9for the
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7 .4 ERROR ESTIMATION 1 24 1PROOF .
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7.5 LAW OF THE ITERATED LOGARITHM 1
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7 .5 LAW OF THE ITERATED LOGARITHM
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7 .6 INFINITE DIVISIBILITY 1 251amo
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7.6 INFINITE DIVISIBILITY 1 253The
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7 .6 INFINITE DIVISIBILITY 1 255Sin
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7 .6 INFINITE DIVISIBILITY 1 257is
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7 .6 INFINITE DIVISIBILITY 1 259Let
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7 .6 INFINITE DIVISIBILITY 1 261ch
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8Random walk8 .1 Zero-or-one lawsIn
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8 .1 ZERO-OR-ONE LAWS 1 2 65Each S2
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8 .1 ZERO-OR-ONE LAWS 1 2 6 71 clea
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8.1 ZERO-OR-ONE LAWS 1 269Applying
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8 .2 BASIC NOTIONS 1 2 7 1have, how
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8 .2 BASIC NOTIONS 1 273Instead of
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8 .2 BASIC NOTIONS 1 275PROOF . The
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8.2 BASIC NOTIONS 1 27 7Theorem 8.2
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8.3 RECURRENCE I 2 7 9DEFINrrION .
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8.3 RECURRENCE 1 28 1by the previou
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8.3 RECURRENCE 1 2 831 A> U ( )Cm n
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8 .3 RECURRENCE 1 285< 2(1 - r) 2 +
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8 .3 RECURRENCE 1 2 8 7is a strictl
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8 .4 FINE STRUCTURE 1 289with the u
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8 .4 FINE STRUCTURE 1 29 1rearrange
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8.4 FINE STRUCTURE 1 293(D) If c,(,
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8.4 FINE STRUCTURE I 2 9 5PROOF . I
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8 .4 FINE STRUCTURE 1 2 9 7for 0 <
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8 .5 CONTINUATION 1 299(necessarily
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8 .5 CONTINUATION 1 30 1and consequ
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8 .5 CONTINUATION 1 3 0 3Lemma .For
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8 .5 CONTINUATION 1 3 0 5PROOF . Le
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Page 341 and 342: 9 .2 CONDITIONAL INDEPENDENCE; MARK
Page 343 and 344: 9.2 CONDITIONAL INDEPENDENCE ; MARK
Page 345 and 346: 9 .2 CONDITIONAL INDEPENDENCE ; MAR
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Page 349 and 350: 9 .3 BASIC PROPERTIES OF SMARTINGAL
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Page 361 and 362: 9 .4 INEQUALITIES AND CONVERGENCE 1
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Page 375 and 376: 9.5 APPLICATIONS 1 361PROOF. Let A
Page 377 and 378: 9 .5 APPLICATIONS 1 363extension of
Page 379 and 380: 9.5 APPLICATIONS 1 365(V)The strong
Page 381 and 382: 9.5 APPLICATIONS 1 3 6 7But it is t
Page 383 and 384: 9 .5 APPLICATIONS 1 369We have ther
Page 385 and 386: 9 .5 APPLICATIONS ( 3712. Suppose t
Page 387 and 388: 9 .5 APPLICATIONS 1 373is due to Mc
Page 389 and 390: Supplement : Measure andIntegralFor
Page 391 and 392: I CONSTRUCTION OF MEASURE 1 377The
Page 393 and 394: 1 CONSTRUCTION OF MEASURE 1 379It f
Page 395 and 396: 2 CHARACTERIZATION OF EXTENSIONS 1
Page 401 and 402: 3 MEASURES IN R 1 387Now we use Def
Page 403 and 404: 3 MEASURES IN R 1 389The right-cont
Page 405 and 406: 3 MEASURES IN R 1 39 1Therefore the
Page 407 and 408: 3 MEASURES IN R 1 393If we intersec
Page 409 and 410: 4 INTEGRAL 1 395has cardinal 2C whi
Page 411 and 412: 4 INTEGRAL 1 397The order of summat
Page 413 and 414: n4 INTEGRAL 1 399If W E A k , then
Page 415 and 416: 4 INTEGRAL 1 401Theorem 9 . Let If,
Page 417 and 418: 4 INTEGRAL I 403inThe three theorem
Page 419 and 420: 4 INTEGRAL 1 405To prove this we ma
Page 421 and 422: 5 APPLICATIONS 1 407Curiously, the
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5 APPLICATIONS 1 409When Uk is gene
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log ~ _ .d~ f~ - dx = log ,5 APPLIC
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General bibliography[The five divis
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IndexAbelian theorem, 292Absolutely
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INDEX 1 417GGambler's ruin, 343Gamb
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INDEX 1 419improper, infinite, 410p
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Course in Probability Theory

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