Course in Probability Theory

More documents

Recommendations

Info

$symbolic math guide : an innovative way of teaching ... - Math.UOC$

298 1 RANDOM WALKlimit theorem, and certainly O(11n) in any event, proposition (C) above willidentify it as the right member of (23) with A = (0, oc) .Now by analogy with (27), replacing (0, cc) by (-oo, 0] and writinga(_oc,o] as 8, we have(28) ({S p} = v lim exp n 1 J'(S,, < 0)rTl n 2n=1Clearly the product of the two exponentials in (27) and (28) is just exp 0 = 1,hence if the limit in (27) were +oc, that in (28) would have to be 0 . Butsince c% (X) = 0 and (`(X2) > 0, we have T(X < 0) > 0, which implies atonce %/'(Sp < 0) > 0 and consequently cf{S f } < 0 . This contradiction provesthat the limits in (27) and (28) must both be finite, and the theorem is proved .Theorem 8 .4 .7 . Suppose that X # 0 and at least one of (,(X+) and (-,''(X- )is finite ; and let a = a ( o , oc ) , , = a (_,, o 1 .(i) If (-r(X) > 0 but may be +oo, then ASa ) =(ii) If t(X) = 0, then (F(Sa ) and o (S p) are both finite if and only if(~, '(X 2 ) < 00 .PROOF. The assertion (i) is a consequence of (18) and Wald's equation(Theorem 5 .5.3 and Exercise 8 of Sec . 5 .5) . The "if' part of assertion (ii)has been proved in the preceding theorem ; indeed we have even "evaluated"(Sa ) . To prove the "only if' part, we apply (11) to both a and ,8 in thepreceding proof and multiply the results together to obtain the remarkableequation :(29) [ 1 - c { . a e tts a}]{l - C`{rIe`rs0}] = exp -- f (t) n = 1 - r f (t) .Setting r = 1, we obtain for t ; 0 :1 - f (t) 1 - c'{eitS-} 1 - (`L{el tSf}t 2 -it +itLetting t ,(, 0, the right member above tends to (-,"ISa}c`'{-S~} by Theorem 6 .4 .2 .Hence the left member has a real and finite limit . This implies (`(X2) < 00 byExercise 1 of Sec . 6 .4 .8 .5 ContinuationOur next study concerns the r .v .'s M n and M defined in (12) and (13) ofSec . 8 .2. It is convenient to introduce a new r.v . L, which is the first time
8 .5 CONTINUATION 1 299(necessarily belonging to N,,O ) that M n is attained :(1) Vn ENO : L,(co) = min{k E N° : Sk(w) = M n ((0)} ;note that Lo = 0 .We shall also use the abbreviation(2)a = a(O,00),8 = a(-00,0]as in Theorems 8 .4.6 and 8 .4 .7 .For each n consider the special permutation below :(3) pn_ l, 2, nn n-1 ,1)'which amounts to a "reversal" of the ordered set of indices in N n . Recallingthat 8 ° p n is the r .v . whose value at w is 6(p n w), and Sk ° pn = Sn - Sn-kfor 1 < k < n, we have{~ ° pn > n } =nn{Sk ° pn > 0}k=1It follows from (8) of Sec . 8 .1 that(4) e'tsn d 7 =n n-1= n {Sn > Sn-k) = n {Sn > Sk } _ {L n = n} .k=1 k=0e` t(s n ° Pn ) dJi = f e'tsn dg') .4>n} f{~ ° p, >n} {Ln=n}Applying (5) and (12) of Sec . 8 .4 to fi and substituting (4), we obtain(5)00 00`Y ,1rn e'tsn d ;? = exp+L_ e'tsn d9' ;n =o {L,=n} n=1 n {Sn>0}applying (5) and (12) of Sec . 8 .4 to{a > n } = {L,, = 0}, we obtaina and substituting the obvious relation(6)00 00• rnfe' ts n d :%T = exp +>-n=0 {L,=O}r'l e't.Sn ,~dJn{sn
Page 2 and 3:
ACOURSE INPROBABILITYTHEORYTHIRD ED
Page 4 and 5:
This book is printed on acid-free p
Page 6 and 7:
vi I CONTENTS4 .3 Vague convergence
Page 8 and 9:
Preface to the third editionIn this
Page 10 and 11:
Preface to the second editionThis e
Page 12 and 13:
Preface to the first editionA mathe
Page 14 and 15:
PREFACE TO THE FIRST EDITION I xv(d
Page 16 and 17:
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
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
2.2 PROBABILITY MEASURES AND THEIR
Page 49 and 50:
3 .1 GENERAL DEFINITIONS 1 3 5of a
Page 51 and 52:
3 .1 GENERAL DEFINITIONS 1 37Specif
Page 53 and 54:
3 .1 GENERAL DEFINITIONS 1 39by (2)
Page 55 and 56:
3 .2 PROPERTIES OF MATHEMATICAL EXP
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
3.2 PROPERTIES OF MATHEMATICAL EXPE
Page 65 and 66:
Page 67 and 68:
3 .3 INDEPENDENCE 1 533.3 Independe
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
Page 87 and 88:
4 .1 VARIOUS MODES OF CONVERGENCE 1
Page 89 and 90:
4 .2 ALMOST SURE CONVERGENCE; BOREL
Page 91 and 92:
Page 93 and 94:
4.2 ALMOST SURE CONVERGENCE; BOREL-
Page 95 and 96:
4 .2 ALMOST SURE CONVERGENCE ; BORE
Page 97 and 98:
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
Page 115 and 116:
Page 117 and 118:
4.5 UNIFORM INTEGRABILITY ; CONVERG
Page 119 and 120:
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
Page 145 and 146:
Page 147 and 148:
5.4 STRONG LAW OF LARGENUMBERS 1 13
Page 149 and 150:
Page 151 and 152:
5.4 STRONG LAW OF LARGENUMBERS 1 13
Page 153 and 154:
is an r cv) as follows co) (o) is c
Page 155 and 156:
5 .5 APPLICATIONS 1 1 4 1PROOF . Su
Page 157 and 158:
5 .5 APPLICATIONS 1 1 43null set Z
Page 159 and 160:
5 .5 APPLICATIONS 1 1 45PROOF.Since
Page 161 and 162:
5 .5 APPLICATIONS 1 1 4 7One should
Page 163 and 164:
5 .5 APPLICATIONS 1 149Smile Borel,
Page 165 and 166:
6.1 GENERAL PROPERTIES; CONVOLUTION
Page 167 and 168:
6.1 GENERAL PROPERTIES ; CONVOLUTIO
Page 169 and 170:
6 .1 GENERAL PROPERTIES ; CONVOLUTI
Page 171 and 172:
CS6 .1 GENERAL PROPERTIES; CONVOLUT
Page 173 and 174:
6 .1 GENERAL PROPERTIES ; CONVOLUTI
Page 175 and 176:
x[ J6 .2 UNIQUENESS AND INVERSION 1
Page 177 and 178:
6 .2 UNIQUENESS AND INVERSION 1 163
Page 179 and 180:
6 .2 UNIQUENESS AND INVERSION 1 165
Page 181 and 182:
I6 .2 UNIQUENESS AND INVERSION 1 1
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
Page 205 and 206:
6 .5 REPRESENTATIVE THEOREMS 1 191(
Page 207 and 208:
6 .5 REPRESENTATIVE THEOREMS I 193w
Page 209 and 210:
6 .5 REPRESENTATIVE THEOREMS I 195E
Page 211 and 212:
6 .6 MULTIDIMENSIONAL CASE; LAPLACE
Page 213 and 214:
6 .6 MULTIDIMENSIONAL CASE ; LAPLAC
Page 215 and 216:
6 .6 MULTIDIMENSIONAL CASE ; LAPLAC
Page 217 and 218:
6 .6 MULTIDIMENSIONAL CASE; LAPLACE
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:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
(D if Sn /s'n does7.3 RAMIFICATIONS
Page 241 and 242:
7.3 RAMIFICATIONS OF THE CENTRAL LI
Page 243 and 244:
7 .3 RAMIFICATIONS OF THE CENTRAL L
Page 245 and 246:
Page 247 and 248:
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
Page 259 and 260:
7 .5 LAW OF THE ITERATED LOGARITHM
Page 261 and 262: 7 .5 LAW OF THE ITERATED LOGARITHM
Page 263 and 264: 7 .5 LAW OF THE ITERATED LOGARITHM
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
Page 275 and 276: 7 .6 INFINITE DIVISIBILITY 1 261ch
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
Page 281 and 282: 8 .1 ZERO-OR-ONE LAWS 1 2 6 71 clea
Page 283 and 284: 8.1 ZERO-OR-ONE LAWS 1 269Applying
Page 285 and 286: 8 .2 BASIC NOTIONS 1 2 7 1have, how
Page 287 and 288: 8 .2 BASIC NOTIONS 1 273Instead of
Page 289 and 290: 8 .2 BASIC NOTIONS 1 275PROOF . The
Page 291 and 292: 8.2 BASIC NOTIONS 1 27 7Theorem 8.2
Page 293 and 294: 8.3 RECURRENCE I 2 7 9DEFINrrION .
Page 295 and 296: 8.3 RECURRENCE 1 28 1by the previou
Page 297 and 298: 8.3 RECURRENCE 1 2 831 A> U ( )Cm n
Page 299 and 300: 8 .3 RECURRENCE 1 285< 2(1 - r) 2 +
Page 301 and 302: 8 .3 RECURRENCE 1 2 8 7is a strictl
Page 303 and 304: 8 .4 FINE STRUCTURE 1 289with the u
Page 305 and 306: 8 .4 FINE STRUCTURE 1 29 1rearrange
Page 307 and 308: 8.4 FINE STRUCTURE 1 293(D) If c,(,
Page 309 and 310: 8.4 FINE STRUCTURE I 2 9 5PROOF . I
Page 311: 8 .4 FINE STRUCTURE 1 2 9 7for 0 <
Page 315 and 316: 8 .5 CONTINUATION 1 30 1and consequ
Page 317 and 318: 8 .5 CONTINUATION 1 3 0 3Lemma .For
Page 319 and 320: 8 .5 CONTINUATION 1 3 0 5PROOF . Le
Page 321 and 322: 8 .5 CONTINUATION 1 307implies JT(M
Page 323 and 324: 8 .5 CONTINUATION 1 309The latter p
Page 325 and 326: 9 .1 BASIC PROPERTIES OF CONDITIONA
Page 337 and 338: 9.2 CONDITIONAL INDEPENDENCE ; MARK
Page 339 and 340: 9 .2 CONDITIONAL INDEPENDENCE; MARK
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
Page 347 and 348: 9 .2 CONDITIONAL INDEPENDENCE ; MAR
Page 349 and 350: 9 .3 BASIC PROPERTIES OF SMARTINGAL
Page 351 and 352: 9.3 BASIC PROPERTIES OF SMARTINGALE
Page 353 and 354: 9.3 BASIC PROPERTIES OF SMARTINGALE
Page 361 and 362: 9 .4 INEQUALITIES AND CONVERGENCE 1
Page 363 and 364:
9 .4 INEQUALITIES AND CONVERGENCE 1
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
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 397 and 398:
Page 399 and 400:
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
Page 423 and 424:
5 APPLICATIONS 1 409When Uk is gene
Page 425 and 426:
log ~ _ .d~ f~ - dx = log ,5 APPLIC
Page 427 and 428:
General bibliography[The five divis
Page 429 and 430:
IndexAbelian theorem, 292Absolutely
Page 431 and 432:
INDEX 1 417GGambler's ruin, 343Gamb
Page 433:
INDEX 1 419improper, infinite, 410p
show all

Course in Probability Theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?