Course in Probability Theory

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88 1 CONVERGENCE CONCEPTSwhich states : The set of all s .p.m .'s is sequentially compact with respect tovague convergence . It is often referred to as "Helly's extraction (or selection)principle" .Theorem 4.3 .3 . Given any sequence of s .p.m .'s, there is a subsequence thatconverges vaguely to an s .p.m .PROOF . Here it is convenient to consider the subdistribution function(s .d .f.) F„ defined as follows :Vx : F n (X) = µn (- 00, X] .If , , is a p.m ., then F n is just its d .f. (see Sec . 2 .2) ; in general F n is increasing,right continuous with F n (-oo) = 0 and Fn (+oo) = An (9 1 ) < 1 .Let D be a countable dense set of R 1 , and let irk, k > 1) be anenumeration of it . The sequence of numbers {F n (rl ), n > 1} is bounded, henceby the Bolzano-Weierstrass theorem there is a subsequence {F lk , k > 1} ofthe given sequence such that the limitlim Flk(rl) = fik-+ o0exists ; clearly 0 < L 1 < 1 . Next, the sequence of numbers {F lk (r2 ), k > 1 } isbounded, hence there is a subsequence {F 2k , k > 1} of {F l k, k > 1} such thatlim F2k(r2) = £2k--*00where 0 < £ 2 < 1 . Since {F 2k } is a subsequence of {FU}, it converges also atrl to f l . Continuing, we obtainF11, F12, . . . , Flk, . . . converging at rl ;F21, F22, . . . , F2k, . . . converging at r l , r2 ;. . . . . . . . . . . . . . . . . . . .Fj 1, Fi2, . . . , Fkk, . . . converging at rl, r2, . . . , rj ;. . . . . . . . . . . . . . . . . . . .Now consider the diagonal sequence {Fkk, k > 11 . We assert that it convergesat every rj , j > 1 . To see this let r j be given . Apart from the first j - 1 terms,the sequence {Fkk, k > 1) is a subsequence of {Fjk, k > 11, which convergesat rj and hence limky 00 Fkk (rj ) = f j, as desired .We have thus proved the existence of an infinite subsequence Ink) and afunction G defined and increasing on D such thatYr E D : lim F nk (r) = G(r) .k a o0From G we define a function F on %il as follows :Yx E ?Rl : F (x) = inf G (r) .x
4 .3 VAGUE CONVERGENCE ( 89By Sec . 1 .1, (vii), F is increasing and right continuous . Let C denote the setof its points of continuity ; C is dense in ?4 1 and we show that(8)Vx E C : lim F„ k (x) = F(x) .k+ ocFor, let x E C and c > 0 be given, there exist r, r', and r" in D such thatr < r' < x < r" and F(r") - F(r) < E . Then we haveF(r)
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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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Page 69 and 70: 3 .3 INDEPENDENCE 1 55PROOF . We gi
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
Page 93 and 94: 4.2 ALMOST SURE CONVERGENCE; BOREL-
Page 95 and 96: 4 .2 ALMOST SURE CONVERGENCE ; BORE
Page 99 and 100: 4 .3 VAGUE CONVERGENCE 1 85In this
Page 101: 4 .3 VAGUE CONVERGENCE 1 87PROOF .
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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Page 121 and 122: 5 .1 SIMPLE LIMIT THEOREMS 1 107eve
Page 123 and 124: 5 .1 SIMPLE LIMIT THEOREMS 1 109and
Page 125 and 126: 5 .1 SIMPLE LIMIT THEOREMS 1 1 1 1r
Page 127 and 128: 5.2 WEAK LAW OF LARGENUMBERS 1 1 1
Page 129 and 130: 5.2 WEAK LAW OF LARGENUMBERS 1 1 1
Page 131 and 132: 5.2 WEAK LAW OF LARGENUMBERS I 1 1
Page 133 and 134: 5 .2 WEAK LAW OF LARGENUMBERS I 1 1
Page 135 and 136: 5 .3 CONVERGENCE OF SERIES 1 12112.
Page 137 and 138: 5 .3 CONVERGENCE OF SERIES 1 123whe
Page 139 and 140: 5 .3 CONVERGENCE OF SERIES 1 1 2 5W
Page 141 and 142: 5 .3 CONVERGENCE OF SERIES 1 127Thi
Page 143 and 144: 5 .4 STRONG LAW OF LARGENUMBERS 1 1
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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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