Probability and Measure, Third Edition (Wiley Series in Probability and Statistics)

✍ Scribed by Patrick Billingsley

• Publisher: Wiley-Interscience
• Year: 1995
• Tongue: English
• Leaves: 608
• Series: Wiley Series in Probability and Statistics
• Edition: 3
• Category: Library

✦ Synopsis

Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.

✦ Table of Contents

PROBABILITY AND MEASURE, 3RD ED.......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 6
Contents......Page 8
The Unit Interval......Page 15
The Weak Law of Large Numbers......Page 19
The Strong Law of Large Numbers......Page 22
Length......Page 25
The Measure Theory of Diophantine Approximation......Page 27
Problems......Page 29
Spaces......Page 31
Classes of Sets......Page 32
Probability Measures......Page 36
Lebesgue Measure on the Unit Interval......Page 39
Sequence Space......Page 41
Constructing σ-Fields......Page 44
Problems......Page 46
3. Existence and Extension......Page 50
Construction of the Extension......Page 51
Uniqueness and the π-λ Theorem......Page 55
Lebesgue Measure on the Unit Interval......Page 57
Completeness......Page 58
Two Impossibility Theorems......Page 59
Problems......Page 60
General Formulas......Page 65
Limit Sets......Page 66
Independent Events......Page 67
Subfields......Page 71
The Borel-Cantelli Lemmas......Page 73
The Zero-—One Law......Page 76
Problems......Page 78
Definition......Page 81
Convergence of Random Variables......Page 84
Independence......Page 85
Existence of Independent Sequences......Page 87
Expected Value......Page 90
Inequalities......Page 94
Problems......Page 95
The Strong Law......Page 99
Bernstein's Theorem......Page 100
A Refinement of the Second Borel-Cantelli Lemma......Page 101
Problems......Page 103
Gambler's Ruin......Page 106
Selection Systems......Page 109
Gambling Policies......Page 112
Bold Play......Page 115
Problems......Page 122
Definitions......Page 125
Higher-Order Transitions......Page 128
An Existence Theorem......Page 129
Transience and Persistence......Page 131
Another Criterion for Persistence......Page 135
Stationary Distributions......Page 138
Exponential Convergence......Page 145
Optimal Stopping......Page 147
Problems......Page 153
9. Large Deviations and the Law of the Iterated Logarithm......Page 159
Moment Generating Functions......Page 160
Large Deviations......Page 162
Chernoff's Theorem......Page 165
The Law of the Iterated Logarithm......Page 167
Problems......Page 171
Classes of Sets......Page 172
Measures......Page 174
Uniqueness......Page 177
Problems......Page 178
Outer Measure......Page 179
Extension......Page 180
An Approximation Theorem......Page 182
Problems......Page 184
Lebesgue Measure......Page 185
Regularity......Page 188
Specifying Measures on the Line......Page 189
Specifying Measures in R k......Page 190
Strange Euclidean Sets......Page 193
Problems......Page 194
Measurable Mappings......Page 196
Mappings into R k......Page 197
Limits and Measurability......Page 198
Transformations of Measures......Page 199
Problems......Page 200
Distribution Functions......Page 201
Exponential Distributions......Page 203
Weak Convergence......Page 204
Convergence of Types......Page 207
Extremal Distributions......Page 209
Problems......Page 211
Definition......Page 213
Nonnegative Functions......Page 215
Uniqueness......Page 217
Problems......Page 218
Equalities and Inequalities......Page 220
Integration to the Limit......Page 222
Integration over Sets......Page 226
Densities......Page 227
Change of Variable......Page 229
Uniform Integrability......Page 230
Complex Functions......Page 232
Problems......Page 233
The Riemann Integral......Page 235
Change of Variable......Page 238
The Lebesgue Integral in R k......Page 239
Problems......Page 242
Product Spaces......Page 245
Product Measure......Page 246
Fubini's Theorem......Page 247
Integration by Parts......Page 250
Products of Higher Order......Page 252
Problems......Page 253
Definitions......Page 255
Completeness and Separability......Page 257
Conjugate Spaces......Page 258
Weak Compactness......Page 260
Some Decision Theory......Page 261
The Space L²......Page 263
An Estimation Problem......Page 265
Problems......Page 266
Random Variables and Vectors......Page 268
Subfields......Page 269
Distributions......Page 270
Multidimensional Distributions......Page 273
Independence......Page 275
Sequences of Random Variables......Page 279
Convolution......Page 280
The Glivenko-Cantelli Theorem......Page 282
Problems......Page 284
Expected Values and Limits......Page 287
Moments......Page 288
Inequalities......Page 290
Independence and Expected Value......Page 291
Moment Generating Functions......Page 292
Problems......Page 294
The Strong Law of Large Numbers......Page 296
The Weak Law and Moment Generating Functions......Page 298
Kolmogorov's Zero-One Law......Page 300
Maximal Inequalities......Page 301
Convergence of Random Series......Page 303
Random Taylor Series......Page 306
Problems......Page 308
The Poisson Process......Page 311
The Poisson Approximation......Page 316
Other Characterizations of the Poisson Process......Page 317
Stochastic Processes......Page 322
Problems......Page 323
24. The Ergodic Theorem......Page 324
Measure-Preserving Transformations......Page 325
Ergodicity......Page 327
Ergodicity of Rotations......Page 330
Proof of the Ergodic Theorem......Page 331
The Continued-Fraction Transformation......Page 333
Diophantine Approximation......Page 338
Problems......Page 339
Definitions......Page 341
Uniform Distribution Modulo 1......Page 342
Convergence in Distribution......Page 343
Convergence in Probability......Page 344
Fundamental Theorems......Page 347
Helly's Theorem......Page 350
Integration to the Limit......Page 352
Problems......Page 353
Moments and Derivatives......Page 356
Independence......Page 359
Inversion and the Uniqueness Theorem......Page 360
The Continuity Theorem......Page 363
Fourier Series......Page 365
Problems......Page 367
Identically Distributed Summands......Page 371
The Lindeberg and Lyapounov Theorems......Page 373
Dependent Variables......Page 377
Problems......Page 382
The Possible Limits......Page 385
Characterizing the Limit......Page 389
Problems......Page 390
The Basic Theorems......Page 392
Characteristic Functions......Page 395
Normal Distributions in R k......Page 397
Problems......Page 399
The Moment Problem......Page 402
Moment Generating Functions......Page 404
Central Limit Theorem by Moments......Page 405
Application to Sampling Theory......Page 406
Application to Number Theory......Page 407
Problems......Page 411
The Fundamental Theorem of Calculus......Page 414
Derivatives of Integrals......Page 416
Singular Functions......Page 421
Integrals of Derivatives......Page 426
Functions of Bounded Variation......Page 429
Problems......Page 430
Additive Set Functions......Page 433
The Hahn Decomposition......Page 434
Absolute Continuity and Singularity......Page 435
The Main Theorem......Page 436
Problems......Page 439
The Discrete Case......Page 441
The General Case......Page 443
Properties of Conditional Probability......Page 450
Difficulties and Curiosities......Page 451
Conditional Probability Distributions......Page 453
Problems......Page 455
Definition......Page 459
Properties of Conditional Expectation......Page 460
Conditional Distributions and Expectations......Page 463
Sufficient Subfields......Page 464
Minimum-Variance Estimation......Page 468
Problems......Page 469
Definition......Page 472
Submartingales......Page 476
Gambling......Page 477
Stopping Times......Page 479
Inequalities......Page 480
Convergence Theorems......Page 482
Applications: Derivatives......Page 484
Likelihood Ratios......Page 485
Reversed Martingales......Page 486
Applications: de Finetti's Theorem......Page 487
A Central Limit Theorem......Page 489
Problems......Page 492
Finite-Dimensional Distributions......Page 496
Product Spaces......Page 498
Kolmogorov's Existence Theorem......Page 500
The Inadequacy of R T......Page 506
A Return to Ergodic Theory......Page 508
Problems......Page 510
Definition......Page 512
Continuity of Paths......Page 514
Measurable Processes......Page 517
Irregularity of Brownian Motion Paths......Page 518
The Strong Markov Property......Page 522
The Reflection Principle......Page 525
Skorohod Embedding......Page 527
Invariance*......Page 534
Problems......Page 537
Definitions......Page 540
Existence Theorems......Page 543
Consequences of Separability......Page 546
Set Theory......Page 550
The Real Line......Page 551
Euclidean k-Space......Page 553
Analysis......Page 554
Infinite Series......Page 556
Convex Functions......Page 558
Some Multivariable Calculus......Page 559
Continued Fractions......Page 561
Section 1......Page 566
Section 2......Page 567
Section 3......Page 569
Section 4......Page 570
Section 5......Page 571
Section 7......Page 572
Section 8......Page 573
Section 9......Page 575
Section 12......Page 576
Section 15......Page 577
Section 16......Page 578
Section 17......Page 579
Section 20......Page 580
Section 22......Page 582
Section 23......Page 583
Section 26......Page 584
Section 27......Page 585
Section 28......Page 586
Section 30......Page 587
Section 31......Page 588
Section 32......Page 589
Section 33......Page 590
Section 34......Page 591
Section 35......Page 592
Section 37......Page 593
Bibliography......Page 595
List of Symbols......Page 599
Index......Page 601
Back Cover......Page 608