Course in Probability Theory

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5 4 1 RANDOM VARIABLE . EXPECTATION . INDEPENDENCEFrom now on when the probability space is fixed, a set in `T will also becalled an event . The events {E j , 1 < j < n } are said to be independent iff theirindicators are independent; this is equivalent to : for any subset { j 1, . . . j f } of11, . . . , n }, we have(4)een E j k. = H?1') (Ej k ) .k=1 k=1Theorem 3 .3 .1 . If {X j , 1 < j < n} are independent r.v .'s and { f j , 1 < j
3 .3 INDEPENDENCE 1 55PROOF . We give two proofs in detail of this important result to illustratethe methods . Cf. the two proofs of (14) in Sec . 3 .2, one indicated there, andone in Exercise 9 of Sec . 3 .2 .First proof. Suppose first that the two r .v.'s X and Y are both discretebelonging respectively to the weighted partitions {A 1 ;c j } and {Mk .;d k } suchthat A j = {X = c j }, M k = { Y = dd . ThusNow we havec`(X) = j ,JT(A 1 ), c`'(Y) = d :%I'(M k ) .jn UMk = U(AjMk)kj,kandX (w)Y(w) = c j dk if Co E A jMk .Hence the r.v . XY is discrete and belongs to the superposed partitions{A jMk ; cjdk} with both the j and the k varying independently of each other .Since X and Y are independent, we have for every j and k :~'(A jMk) = P(X = c1 ; Y = d k ) = ?P(X = c j )~'(Y = dk) _ 'T(Aj)^Mk) ;and consequently by definition (1) of Sec . 3 .2 :c`(XY) = jdk'(AjMk) = c1 :Y(A1) dk'J'(Mk)j .kj= c (X)c`(Y) .Thus (5) is true in this case .Now let X and Y be arbitrary positive r .v .'s with finite expectations . Then,according to the discussion at the beginning of Sec . 3 .2, there are discreter .v .'s X,,, and Y,,, such that ((X,,,) T c`(X) and (`(Y, n ) T C "(Y) . Furthermore,for each m, X,n and Y,,, are independent . Note that for the independence ofdiscrete r .v .'s it is sufficient to verify the relation (1) when the B j 's are theirpossible values and "E" is replaced by "=" (why ?). Here we haven n' n n+1 n' n'+l~~p Xm - 2rn' Yn, = 2177 { 2rn -X < 2m , 2rn < Y
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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
Page 18 and 19: 1 .1 MONOTONE FUNCTIONS 1 3Example
Page 20 and 21: 1 .1 MONOTONE FUNCTIONS 1 5havef I
Page 22 and 23: 8 1 DISTRIBUTION FUNCTIONb ithe siz
Page 24 and 25: 10 1 DISTRIBUTION FUNCTIONF2 - F ;
Page 26 and 27: 12 ( DISTRIBUTION FUNCTIONThe next
Page 28 and 29: 14 1 DISTRIBUTION FUNCTIONbecomes o
Page 30: Measure theory2 .1 Classes of setsL
Page 33 and 34: 2 .1 CLASSES OF SETS 1 19and so = 6
Page 35 and 36: 2.2 PROBABILITY MEASURES AND THEIR
Page 37 and 38: 2 .2 PROBABILITY MEASURES AND THEIR
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Page 49 and 50: 3 .1 GENERAL DEFINITIONS 1 3 5of a
Page 51 and 52: 3 .1 GENERAL DEFINITIONS 1 37Specif
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Page 55 and 56: 3 .2 PROPERTIES OF MATHEMATICAL EXP
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Page 67: 3 .3 INDEPENDENCE 1 533.3 Independe
Page 71 and 72: 3 .3 INDEPENDENCE 1 57Corollary . I
Page 73 and 74: 3 .3 INDEPENDENCE 1 59Then we haven
Page 75 and 76: 3 .3 INDEPENDENCE 1 61in which ther
Page 77 and 78: 3 .3 INDEPENDENCE 1 63We have thus
Page 79 and 80: 3 .3 INDEPENDENCE I 65EXERCISES1 .
Page 81 and 82: 3 .3 INDEPENDENCE I 67If do : {E~n
Page 83 and 84: 4 .1 VARIOUS MODES OF CONVERGENCE 1
Page 85 and 86: 4 .1 VARIOUS MODES OF CONVERGENCE I
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Page 89 and 90: 4 .2 ALMOST SURE CONVERGENCE; BOREL
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Page 95 and 96: 4 .2 ALMOST SURE CONVERGENCE ; BORE
Page 99 and 100: 4 .3 VAGUE CONVERGENCE 1 85In this
Page 101 and 102: 4 .3 VAGUE CONVERGENCE 1 87PROOF .
Page 103 and 104: 4 .3 VAGUE CONVERGENCE ( 89By Sec .
Page 105 and 106: 4.4 CONTINUATION 1 918. If g„ and
Page 107 and 108: 4 .4 CONTINUATION 1 93that is linea
Page 109 and 110: 4.4 CONTINUATION 1 95A family of p
Page 111 and 112: 4 .4 CONTINUATION 1 97if X„ -+ X
Page 113 and 114: 4 .5 UNIFORM INTEGRABILITY ; CONVER
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4 .5 UNIFORM INTEGRABILITY ; CONVER
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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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8 .5 CONTINUATION 1 307implies JT(M
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8 .5 CONTINUATION 1 309The latter p
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9 .1 BASIC PROPERTIES OF CONDITIONA
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9.2 CONDITIONAL INDEPENDENCE ; MARK
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9 .2 CONDITIONAL INDEPENDENCE; MARK
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9 .2 CONDITIONAL INDEPENDENCE; MARK
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9.2 CONDITIONAL INDEPENDENCE ; MARK
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9 .2 CONDITIONAL INDEPENDENCE ; MAR
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9 .2 CONDITIONAL INDEPENDENCE ; MAR
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9 .3 BASIC PROPERTIES OF SMARTINGAL
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9.3 BASIC PROPERTIES OF SMARTINGALE
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9.3 BASIC PROPERTIES OF SMARTINGALE
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9 .4 INEQUALITIES AND CONVERGENCE 1
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9.5 APPLICATIONS 1 361PROOF. Let A
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9 .5 APPLICATIONS 1 363extension of
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9.5 APPLICATIONS 1 365(V)The strong
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9.5 APPLICATIONS 1 3 6 7But it is t
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9 .5 APPLICATIONS 1 369We have ther
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9 .5 APPLICATIONS ( 3712. Suppose t
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9 .5 APPLICATIONS 1 373is due to Mc
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Supplement : Measure andIntegralFor
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I CONSTRUCTION OF MEASURE 1 377The
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1 CONSTRUCTION OF MEASURE 1 379It f
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2 CHARACTERIZATION OF EXTENSIONS 1
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3 MEASURES IN R 1 387Now we use Def
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3 MEASURES IN R 1 389The right-cont
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3 MEASURES IN R 1 39 1Therefore the
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3 MEASURES IN R 1 393If we intersec
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4 INTEGRAL 1 395has cardinal 2C whi
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4 INTEGRAL 1 397The order of summat
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n4 INTEGRAL 1 399If W E A k , then
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4 INTEGRAL 1 401Theorem 9 . Let If,
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4 INTEGRAL I 403inThe three theorem
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4 INTEGRAL 1 405To prove this we ma
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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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