Probability: Theory and Examples

by Rick Durrett

• ISBN13: 9780521765398
• ISBN10: 0521765390
• Edition: 4th
• Format: Hardcover
• Copyright: 2010-08-30
• Publisher: Cambridge University Press

Summary

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Author Biography

Rick Durrett received his Ph.D. in operations research from Stanford University in 1976. After nine years at UCLA and twenty-five at Cornell University, he moved to Duke University in 2010, where he is a professor of mathematics. He is the author of 8 books and more than 170 journal articles on a wide variety of topics, and he has supervised more than 40 Ph.D. students. He is a member of the National Academy of Science and the American Academy of Arts and Sciences and a Fellow of the Institute of Mathematical Statistics.

Table of Contents

Preface	p. ix
Measure Theory	p. 1
Probability Spaces	p. 1
Distributions	p. 9
Random Variables	p. 14
Integration	p. 17
Properties of the Integral	p. 23
Expected Value	p. 27
Inequalities	p. 27
Integration to the Limit	p. 29
Computing Expected Values	p. 30
Product Measures, Fubini's Theorem	p. 36
Laws of Large Numbers	p. 41
Independence	p. 41
Sufficient Conditions for Independence	p. 43
Independence, Distribution, and Expectation	p. 45
Sums of Independent Random Variables	p. 47
Constructing Independent Random Variables	p. 50
Weak Laws of Large Numbers	p. 53
L2 Weak Laws	p. 53
Triangular Arrays	p. 56
Truncation	p. 59
Borel-Cantelli Lemmas	p. 64
Strong Law of Large Numbers	p. 73
Convergence of Random Series*	p. 78
Rates of Convergence	p. 82
Infinite Mean	p. 84
Large Deviations*	p. 86
Central Limit Theorems	p. 94
The De Moivre-Laplace Theorem	p. 94
Weak Convergence	p. 97
Examples	p. 97
Theory	p. 100
Characteristic Functions	p. 106
Definition, Inversion Formula	p. 106
Weak Convergence	p. 112
Moments and Derivatives	p. 114
Polya's Criterion*	p. 118
The Moment Problem*	p. 120
Central Limit Theorems	p. 124
i.i.d. Sequences	p. 124
Triangular Arrays	p. 129
Prime Divisors (Erdös-Kac)*	p. 133
Rates of Convergence (Berry-Esseen)*	p. 137
Local Limit Theorems*	p. 141
Poisson Convergence	p. 146
The Basic Limit Theorem	p. 146
Two Examples with Dependence	p. 151
Poisson Processes	p. 154
Stable Laws*	p. 158
Infinitely Divisible Distributions*	p. 169
Limit Theorems in Rd	p. 172
Random Walks	p. 179
Stopping Times	p. 179
Recurrence	p. 189
Visits to 0, Arcsine Laws*	p. 201
Renewal Theory*	p. 208
Martingales	p. 221
Conditional Expectation	p. 221
Examples	p. 223
Properties	p. 226
Regular Conditional Probabilities*	p. 230
Martingales, Almost Sure Convergence	p. 232
Examples	p. 239
Bounded Increments	p. 239
Polya's Urn Scheme	p. 241
Radon-Nikodym Derivatives	p. 242
Branching Processes	p. 245
Doob's Inequality, Convergence in Lp	p. 249
Square Integrable Martingales*	p. 254
Uniform Integrability, Convergence in L1	p. 258
Backwards Martingales	p. 264
Optional Stopping Theorems	p. 269
Markov Chains	p. 274
Definitions	p. 274
Examples	p. 277
Extensions of the Markov Property	p. 282
Recurrence and Transience	p. 288
Stationary Measures	p. 296
Asymptotic Behavior	p. 307
Periodicity, Tail ¿-field*	p. 314
General State Space*	p. 318
Recurrence and Transience	p. 322
Stationary Measures	p. 323
Convergence Theorem	p. 324
GI/G/1 Queue	p. 325
Ergodic Theorems	p. 328
Definitions and Examples	p. 328
Birkhoff's Ergodic Theorem	p. 333
Recurrence	p. 338
A Subadditive Ergodic Theorem*	p. 342
Applications*	p. 347
Brownian Motion	p. 353
Definition and Construction	p. 353
Markov Property, Blumenthal's 0-1 Law	p. 359
Stopping Times, Strong Markov Property	p. 365
Path Properties	p. 370
Zeros of Brownian Motion	p. 370
Hitting Times	p. 371
Lévy's Modulus of Continuity	p. 375
Martingales	p. 376
Multidimensional Brownian Motion	p. 380
Donsker's Theorem	p. 382
Empirical Distributions, Brownian Bridge	p. 391
Laws of the Iterated Logarithm*	p. 396
Appendix A: Measure Theory Details	p. 401
Carathéodory's Extension Theorem	p. 401
Which Sets Are Measurable?	p. 407
Kolmogorov's Extension Theorem	p. 410
Radon-Nikodym Theorem	p. 412
Differentiating under the Integral	p. 416
References	p. 419
Index	p. 425
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