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# Probability: Theory and Examples

**by**Rick Durrett

4th

### 9780521765398

0521765390

Hardcover

8/30/2010

Cambridge University Press

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## 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 |

L^{2} 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 R^{d} | 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 L^{p} | p. 249 |

Square Integrable Martingales^{*} | p. 254 |

Uniform Integrability, Convergence in L^{1} | 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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