Course in Probability Theory

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216 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSand consequentlya2 2(X"1) _ x2 dF ni (x) - C (X;,~ )2fix> f _f }x 2 dFn ~ (x) .I
7.2 LINDEBERG-FELLER THEOREM 1 217We apply the lemma to (1) as follows . For each m > 1, we havelim m f x 2 d F n jn(x) = 0 .+oo j1 = IXI>1/mIt follows that there exists a sequence {>7n }decreasing to 0 such thatfj=1XI > 1/nx 2 dF n j (x) --)~ 0 .Now we can go back and modify the definition in (3) by replacing 77 withi) n . As indicated above, the cited corollary becomes applicable and yields theconvergence of [S ;, - ~~(S;,)]/sn in dist . to (Y, hence also that of Sn/s' asremarked .Finally we must go from Sn to S n . The idea is similar to Theorem 5 .2 .1but simpler . Observe that, for the modified Xn j in (3) with 27 replaced by 17n,we have~P{Sn :A Sn } -
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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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Page 183 and 184: 6 .3 CONVERGENCE THEOREMS I 169For
Page 185 and 186: 6 .3 CONVERGENCE THEOREMS 1 171ther
Page 187 and 188: 6.3 CONVERGENCE THEOREMS 1 173metri
Page 189 and 190: 6 .4 SIMPLE APPLICATIONS I 175b > a
Page 191 and 192: 6 .4 SIMPLE APPLICATIONS I 177k-1 j
Page 193 and 194: 6 .4 SIMPLE APPLICATIONS 1 179Theor
Page 195 and 196: 6 .4 SIMPLE APPLICATIONS 1 181then
Page 197 and 198: 6 .4 SIMPLE APPLICATIONS 1 183We en
Page 199 and 200: 6 .4 SIMPLE APPLICATIONS 1 185limit
Page 201 and 202: 6 .5 REPRESENTATIVE THEOREMS 1 187c
Page 203 and 204: l6 .5 REPRESENTATIVE THEOREMS 1 189
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Page 207 and 208: 6 .5 REPRESENTATIVE THEOREMS I 193w
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Page 211 and 212: 6 .6 MULTIDIMENSIONAL CASE; LAPLACE
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Page 219 and 220: 7Central limit theorem andits ramif
Page 221 and 222: 7.1 LIAPOUNOV'S THEOREM l 207are "n
Page 223 and 224: 7.1 LIAPOUNOV'S THEOREM 1 209the sa
Page 225 and 226: (D7.1 LIAPOUNOV'S THEOREM 1 211If1,
Page 227 and 228: 7 .1 LIAPOUNOV'S THEOREM I 213and p
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Page 239 and 240: (D if Sn /s'n does7.3 RAMIFICATIONS
Page 241 and 242: 7.3 RAMIFICATIONS OF THE CENTRAL LI
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Page 249 and 250: 7 .4 ERROR ESTIMATION 1 2 35as n -~
Page 251 and 252: 22x2x27 .4 ERROR ESTIMATION 1 23 7b
Page 253 and 254: 7 .4 ERROR ESTIMATION l 23 9for the
Page 255 and 256: 7 .4 ERROR ESTIMATION 1 24 1PROOF .
Page 257 and 258: 7.5 LAW OF THE ITERATED LOGARITHM 1
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Page 265 and 266: 7 .6 INFINITE DIVISIBILITY 1 251amo
Page 267 and 268: 7.6 INFINITE DIVISIBILITY 1 253The
Page 269 and 270: 7 .6 INFINITE DIVISIBILITY 1 255Sin
Page 271 and 272: 7 .6 INFINITE DIVISIBILITY 1 257is
Page 273 and 274: 7 .6 INFINITE DIVISIBILITY 1 259Let
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Page 277 and 278: 8Random walk8 .1 Zero-or-one lawsIn
Page 279 and 280: 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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Course in Probability Theory

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