Course in Probability Theory

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222 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSHence Lindeberg's condition is satisfied for the double arrayX,,;1< j
7 .2 LINDEBERG-FELLER THEOREM 1 223*7. Find an example where Lindeberg's condition is satisfied butLiapounov's is not for any 8 > 0 .In Exercises 8 to 10 below {X j , j > 1 } is a sequence of independent r .v .'s .8. For each j let X j have the uniform distribution in [-j, j] . Show thatLindeberg's condition is satisfied and state the resulting central limit theorem .9. Let X,. be defined as follows for some a > 1 :1fja , with probability 6 j2(a_1) each ;X~ - 10, with probability 1 - 3 j2(a-1)Prove that Lindeberg's condition is satisfied if and only if a < 3/2 .* 10 . It is important to realize that the failure of Lindeberg's conditionmeans only the failure of either (i) or (ii) in Theorem 7 .2 .1 with the specifiedconstants s, A central limit theorem may well hold with a different sequenceof constants . Letwith probability each ;12j 2with probability 12 each ;with probability 1 - 1 - 16 6j 2Prove that Lindeberg's condition is not satisfied . Nonetheless if we take b, _11 3/18, then S„ /b„ converges in d ist . to (D . The point is that abnormally largevalues may not count! [HINT : Truncate out the abnormal value .]11 . Prove that fo x 2 d F (x) < oo implies the condition (11), but notvice versa .* 12 . The following combinatorial problem is similar to that of the numberof inversions . Let Q and '~P be as in the example in the text . It is standardknowledge that each permutation1a,2a2can be uniquely decomposed into the product of cycles, as follows . Considerthe permutation as a mapping n from the set (1, . . ., n) onto itself suchthat 7r(j) = a j . Beginning with 1 and applying the mapping successively,1 --* 7r (1) - 2r 2 (1) -* . . ., until the first k such that rr k (1) = 1 . Thus(1, 7r(l), 7r2(1), . . . , 7rk-1(1))
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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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Page 189 and 190: 6 .4 SIMPLE APPLICATIONS I 175b > a
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Page 193 and 194: 6 .4 SIMPLE APPLICATIONS 1 179Theor
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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 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
Page 229 and 230: 7 .2 LINDEBERG-FELLER THEOREM 1 215
Page 231 and 232: 7.2 LINDEBERG-FELLER THEOREM 1 217W
Page 233 and 234: 7 .2 LINDEBERG-FELLER THEOREM 1 219
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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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Page 283 and 284: 8.1 ZERO-OR-ONE LAWS 1 269Applying
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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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