Course in Probability Theory

More documents

Recommendations

Info

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

4 1 8 1 INDEXMartingale, 335, see also Submartingale,Supermartingaledifference sequence, 337Krickeberg's decomposition, 358of conditional expectations, 338stopped, 341Maximum of S,,, 228, 299Maximum of submartingale, 346, 362M .C ., see Monotone classm-dependent, 224Mean, 49Measurable with respect to 3 , 35MeasureCompletion, 384Extension, 377, 380a-finite, 381Transfinite construction, 386Uniqueness, 381Measure space, 381discrete, 386Median, 117Method of moments, 103, 178Metricfor ch .f.'s, 172for convergence in pr ., 71for d .f.'s, 98for measure space, 47for p.m .'s, 172Minimal B .F ., 18Minkowski's inequality, 50Moment, 49Moment problem, 103Monotone class theorems, 19Monotone convergence theorem, 44, 401Monotone property of measure, 21Multinomial distribution, 181n-dimensional distribution, 53Negligibility, 206Non-measurable set, 395Normal d .f., 104characterization of, 180, 182convergence to, see Central limit theoremkernel, 157positive, 104, 232Normal number, 109, 112Norming, 206Null set, 30, 383Number of positive terms, 302Number of zeros, 234NOptional r.v ., 259, 338Optional sampling, 338, 346Order statistics, 147Orthogonal, 107Outcome, 57Outer measure, 28, 375, 377-378Pairwise independence, 53Partition, 40Permutable, 266p.m ., see Probability measurePoisson d .f., 194Poisson process, 142P61ya criterion, 191P61ya distribution, 158Positive definite, 187Positive normal, see NormalPost-a, 259Potential, 359, 361Pre-a, 259Probability distribution, 36on circle, 107Probability measure, 20of d.f., 30Probability space, triple, 23Product B .F . measure, space, 58Projection, 64Queuing theory, 307Radon-Nikodym theorem, 313, 369Random variable, 34, see also Discrete,integrable, simple, symmetric randomvariableRandom vector, 37Random walk, 271recurrent, 284Recurrence principle, 360Recurrent value, 278Reflection principle, 232Remainder term, 235Remote event, field, 264Renewal theory, 142Riemann integral, 4090PQR
INDEX 1 419improper, infinite, 410primitive, 411Riemann-Lebesgue lemma, 167Riemann-Stieltjes integral, 198Riemann zeta function, 259Riesz decomposition, 359Right continuity, 5r .v ., see Random variableSample function, 141Sample space, 23discrete, 23s .d .f., see Subdistribution functionSecond-order stochastic process, 191Sequentially vaguely compact, 88Set of convergence, 79Shift, 265optional, 273Simple random variable, 40Singleton, 16Singular d .f., 12measure, 31Singular part of d.f., 12Smartingale, 335closeable, 359Span, 184Spectral distribution, 191St . Petersburg paradox, 120, 373Spitzer's identity, 299Squared variation, 367Stable distribution, 193Standard deviation, 49Stationary independent process, 267Stationary transition probabilities, 331Step function, 92Stochastic process, with discrete parameter,263Stone -Weierstrass theorem, 92, 198Stopped (super) martingale, 340-341Stopping time, see Optional samplingSubdistribution function, 88Submartingale, 335convergence theorems for, 350convex transformation of, 336Doob decomposition, 337inequalities, 346Subprobability measure, 85Subsequences, method of, 109Superharmonic function, 361SSupermartingale, 335optional sampling, 340Support of d.£, 10Support of p.m ., 32Symmetric difference (o), 16Symmetric random variable, 165Symmetric stable law, 193, 250Symmetrization, 155System theorem, see Gambling system,SubmartingaleTauberian theorem, 292Taylor's series, 177Three series theorem, 125Tight, 95Topological measure space, 28Trace, 23Transition probability function, 331Trial, 57Truncation, 115Types of d .f., 184Uncorrelated, 107Uniform distribution, 30Uniform integrability, 99Uniqueness theoremfor ch .f., 160, 168for Laplace transform, 199, 201for measure, 30Upcrossing inequality, 348TUVVague convergence, 85, 92general criterion, 96Variance, 49WWald's equation, 144, 345Weak convergence, 73Weierstrass theorem, 145Wiener-Hopf technique, 291Zero-or-one lawHewitt-Savage, 268Kolmogorov, 267L6vy, 357Z
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:
Page 263 and 264:
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 and 312:
8 .4 FINE STRUCTURE 1 2 9 7for 0 <
Page 313 and 314:
8 .5 CONTINUATION 1 299(necessarily
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 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
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 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
9 .4 INEQUALITIES AND CONVERGENCE 1
Page 363 and 364:
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 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: INDEX 1 417GGambler's ruin, 343Gamb
show all

Course in Probability Theory

Create successful ePaper yourself

Delete template?

Save as template?