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Probability, Fourth Edition

Book Description

This classic introduction to probability theory for beginning graduate students covers 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. The new edition begins with a short chapter on measure theory to orient readers new to the subject.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. 1 Measure Theory
    1. 1.1 Probability Spaces
    2. 1.2 Distributions
    3. 1.3 Random Variables
    4. 1.4 Integration
    5. 1.5 Properties of the Integral
    6. 1.6 Expected Value
      1. 1.6.1 Inequalities
      2. 1.6.2 Integration to the Limit
      3. 1.6.3 Computing Expected Values
    7. 1.7 Product Measures, Fubini’s Theorem
  7. 2 Laws of Large Numbers
    1. 2.1 Independence
      1. 2.1.1 Sufficient Conditions for Independence
      2. 2.1.2 Independence, Distribution, and Expectation
      3. 2.1.3 Sums of Independent Random Variables
      4. 2.1.4 Constructing Independent Random Variables
    2. 2.2 Weak Laws of Large Numbers
      1. 2.2.1 L[sup(2)] Weak Laws
      2. 2.2.2 Triangular Arrays
      3. 2.2.3 Truncation
    3. 2.3 Borel-Cantelli Lemmas
    4. 2.4 Strong Law of Large Numbers
    5. 2.5 Convergence of Random Series*
      1. 2.5.1 Rates of Convergence
      2. 2.5.2 Infinite Mean
    6. 2.6 Large Deviations*
  8. 3 Central Limit Theorems
    1. 3.1 The De Moivre-Laplace Theorem
    2. 3.2 Weak Convergence
      1. 3.2.1 Examples
      2. 3.2.2 Theory
    3. 3.3 Characteristic Functions
      1. 3.3.1 Definition, Inversion Formula
      2. 3.3.2 Weak Convergence
      3. 3.3.3 Moments and Derivatives
      4. 3.3.4 Polya’s Criterion*
      5. 3.3.5 The Moment Problem*
    4. 3.4 Central Limit Theorems
      1. 3.4.1 i.i.d. Sequences
      2. 3.4.2 Triangular Arrays
      3. 3.4.3 Prime Divisors (Erdös-Kac)*
      4. 3.4.4 Rates of Convergence (Berry-Esseen)*
    5. 3.5 Local Limit Theorems*
    6. 3.6 Poisson Convergence
      1. 3.6.1 The Basic Limit Theorem
      2. 3.6.2 Two Examples with Dependence
      3. 3.6.3 Poisson Processes
    7. 3.7 Stable Laws*
    8. 3.8 Infinitely Divisible Distributions*
    9. 3.9 Limit Theorems in R[sup(d)]
  9. 4 Random Walks
    1. 4.1 Stopping Times
    2. 4.2 Recurrence
    3. 4.3 Visits to 0, Arcsine Laws*
    4. 4.4 Renewal Theory*
  10. 5 Martingales
    1. 5.1 Conditional Expectation
      1. 5.1.1 Examples
      2. 5.1.2 Properties
      3. 5.1.3 Regular Conditional Probabilities*
    2. 5.2 Martingales, Almost Sure Convergence
    3. 5.3 Examples
      1. 5.3.1 Bounded Increments
      2. 5.3.2 Polya’s Urn Scheme
      3. 5.3.3 Radon-Nikodym Derivatives
      4. 5.3.4 Branching Processes
    4. 5.4 Doob’s Inequality, Convergence in L[sup(p)]
      1. 5.4.1 Square Integrable Martingales*
    5. 5.5 Uniform Integrability, Convergence in L[sup(1)]
    6. 5.6 Backwards Martingales
    7. 5.7 Optional Stopping Theorems
  11. 6 Markov Chains
    1. 6.1 Definitions
    2. 6.2 Examples
    3. 6.3 Extensions of the Markov Property
    4. 6.4 Recurrence and Transience
    5. 6.5 Stationary Measures
    6. 6.6 Asymptotic Behavior
    7. 6.7 Periodicity, Tail σ-field*
    8. 6.8 General State Space*
      1. 6.8.1 Recurrence and Transience
      2. 6.8.2 Stationary Measures
      3. 6.8.3 Convergence Theorem
      4. 6.8.4 GI/G/1 Queue
  12. 7 Ergodic Theorems
    1. 7.1 Definitions and Examples
    2. 7.2 Birkhoff’s Ergodic Theorem
    3. 7.3 Recurrence
    4. 7.4 A Subadditive Ergodic Theorem*
    5. 7.5 Applications*
  13. 8 Brownian Motion
    1. 8.1 Definition and Construction
    2. 8.2 Markov Property, Blumenthal’s 0-1 Law
    3. 8.3 Stopping Times, Strong Markov Property
    4. 8.4 Path Properties
      1. 8.4.1 Zeros of Brownian Motion
      2. 8.4.2 Hitting Times
      3. 8.4.3 Lévy’s Modulus of Continuity
    5. 8.5 Martingales
      1. 8.5.1 Multidimensional Brownian Motion
    6. 8.6 Donsker’s Theorem
    7. 8.7 Empirical Distributions, Brownian Bridge
    8. 8.8 Laws of the Iterated Logarithm*
  14. Appendix A: Measure Theory Details
    1. A.1 Carathéodory’s Extension Theorem
    2. A.2 Which Sets Are Measurable?
    3. A.3 Kolmogorov’s Extension Theorem
    4. A.4 Radon-Nikodym Theorem
    5. A.5 Differentiating under the Integral
  15. References
  16. Index