Probability: Theory and Examples / Edition 2

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Modern and measure-theory based, this text is intended primarily for the first-year graduate course in probability theory.

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Editorial Reviews

A text for a one-year graduate course. Assumes familiarity with measure theory but includes proofs of the necessary facts in an appendix. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780534243180
  • Publisher: Cengage Learning
  • Publication date: 9/28/1995
  • Edition description: Older Edition
  • Edition number: 2
  • Pages: 528
  • Product dimensions: 6.61 (w) x 9.57 (h) x 1.01 (d)

Table of Contents

INTRODUCTORY LECTURE 1. LAWS OF LARGE NUMBERS Basic Definitions / Random Variables / Expected Value / Independence / Weak Laws of Large Numbers / Borel-Cantelli Lemmas / Strong Laws of Large Numbers / Convergence of Random Series / Large Deviations 2. CENTRAL LIMIT THEOREMS The De Moivre-Laplace Theorem / Weak Convergence / Characteristic Functions / Central Limit Theorems / Local Limit Theorems / Poisson Convergence / Stable Laws / Infinitely Divisible Distributions / Limit Theorems in R^d 3. RANDOM WALKS Stopping Times / Recurrence / Visits to 0, Arcsine Laws / Renewal Theory 4. MARTINGALES Conditional Expectation / Martingales, Almost Sure Convergence / Examples / Doob'''s Inequality / L^p Convergence / Uniform Integrability, Convergence in L^1 / Backwards Martingales / Optional Stopping Theorems 5. MARKOV CHAINS Definitions and Examples / Extensions of the Markov Property / Recurrence and Transience / Stationary Measures / Asymptotic Behavior / General State Space 6. ERGODIC THEOREMS Definitions and Examples / Birkhoff'''s Ergodic Theorem / Recurrence / Mixing / Entropy / A Subadditive Ergodic Theorem / Applications 7. BROWNIAN MOTION Definition and Construction / Markov Property, Blumenthal''s 0-1 Law / Stopping Times, Strong Markov Property / Maxima and Zeros / Martingales / Donsker'''s Theorem / CLT'''s for Dependent Variables / Empirical Distributions, Brownian Bridge / Laws of the Iterated Logarithm / APPENDIX: MEASURE THEORY / Lebesgue-Stieljes Measures / Caratheodary'''s Extension Theorem / Completion, etc. / Integration / Properties of the Integral / Product Measures, Fubini'''s Theorem / Kolmogorov'''s Extension Theorem / Radon-Nikodym Theorem / Differentiating Under the Integral / REFERENCES / NOTATION / NORMAL TABLE / INDEX

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