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Probability : Theory and Examples



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Cambridge University Press
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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

Prefacep. ix
Measure Theoryp. 1
Probability Spacesp. 1
Distributionsp. 9
Random Variablesp. 14
Integrationp. 17
Properties of the Integralp. 23
Expected Valuep. 27
Inequalitiesp. 27
Integration to the Limitp. 29
Computing Expected Valuesp. 30
Product Measures, Fubini's Theoremp. 36
Laws of Large Numbersp. 41
Independencep. 41
Sufficient Conditions for Independencep. 43
Independence, Distribution, and Expectationp. 45
Sums of Independent Random Variablesp. 47
Constructing Independent Random Variablesp. 50
Weak Laws of Large Numbersp. 53
L2 Weak Lawsp. 53
Triangular Arraysp. 56
Truncationp. 59
Borel-Cantelli Lemmasp. 64
Strong Law of Large Numbersp. 73
Convergence of Random Series*p. 78
Rates of Convergencep. 82
Infinite Meanp. 84
Large Deviations*p. 86
Central Limit Theoremsp. 94
The De Moivre-Laplace Theoremp. 94
Weak Convergencep. 97
Examplesp. 97
Theoryp. 100
Characteristic Functionsp. 106
Definition, Inversion Formulap. 106
Weak Convergencep. 112
Moments and Derivativesp. 114
Polya's Criterion*p. 118
The Moment Problem*p. 120
Central Limit Theoremsp. 124
i.i.d. Sequencesp. 124
Triangular Arraysp. 129
Prime Divisors (Erdös-Kac)*p. 133
Rates of Convergence (Berry-Esseen)*p. 137
Local Limit Theorems*p. 141
Poisson Convergencep. 146
The Basic Limit Theoremp. 146
Two Examples with Dependencep. 151
Poisson Processesp. 154
Stable Laws*p. 158
Infinitely Divisible Distributions*p. 169
Limit Theorems in Rdp. 172
Random Walksp. 179
Stopping Timesp. 179
Recurrencep. 189
Visits to 0, Arcsine Laws*p. 201
Renewal Theory*p. 208
Martingalesp. 221
Conditional Expectationp. 221
Examplesp. 223
Propertiesp. 226
Regular Conditional Probabilities*p. 230
Martingales, Almost Sure Convergencep. 232
Examplesp. 239
Bounded Incrementsp. 239
Polya's Urn Schemep. 241
Radon-Nikodym Derivativesp. 242
Branching Processesp. 245
Doob's Inequality, Convergence in Lpp. 249
Square Integrable Martingales*p. 254
Uniform Integrability, Convergence in L1p. 258
Backwards Martingalesp. 264
Optional Stopping Theoremsp. 269
Markov Chainsp. 274
Definitionsp. 274
Examplesp. 277
Extensions of the Markov Propertyp. 282
Recurrence and Transiencep. 288
Stationary Measuresp. 296
Asymptotic Behaviorp. 307
Periodicity, Tail ¿-field*p. 314
General State Space*p. 318
Recurrence and Transiencep. 322
Stationary Measuresp. 323
Convergence Theoremp. 324
GI/G/1 Queuep. 325
Ergodic Theoremsp. 328
Definitions and Examplesp. 328
Birkhoff's Ergodic Theoremp. 333
Recurrencep. 338
A Subadditive Ergodic Theorem*p. 342
Applications*p. 347
Brownian Motionp. 353
Definition and Constructionp. 353
Markov Property, Blumenthal's 0-1 Lawp. 359
Stopping Times, Strong Markov Propertyp. 365
Path Propertiesp. 370
Zeros of Brownian Motionp. 370
Hitting Timesp. 371
Lévy's Modulus of Continuityp. 375
Martingalesp. 376
Multidimensional Brownian Motionp. 380
Donsker's Theoremp. 382
Empirical Distributions, Brownian Bridgep. 391
Laws of the Iterated Logarithm*p. 396
Appendix A: Measure Theory Detailsp. 401
Carathéodory's Extension Theoremp. 401
Which Sets Are Measurable?p. 407
Kolmogorov's Extension Theoremp. 410
Radon-Nikodym Theoremp. 412
Differentiating under the Integralp. 416
Referencesp. 419
Indexp. 425
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