An Introduction to Probability Theory and Its Applications, Volume 2, 2nd Edition

Chapter I The Exponential and the Uniform Densities

1. Introduction

2. Densities. Convolutions

3. The Exponential Density

4. Waiting Time Paradoxes. The Poisson Process

5. The Persistence of Bad Luck

6. Waiting Times and Order Statistics

7. The Uniform Distribution

8. Random Splittings

9. Convolutions and Covering Theorems

10. Random Directions

11. The Use of Lebesgue Measure

12. Empirical Distributions

13. Problems for Solution

Chapter II Special Densities. Randomization

1. Notations and Conventions

2. Gamma Distributions

3. Related Distributions of Statistics

4. Some Common Densities

5. Randomization and Mixtures

6. Discrete Distributions

7. Bessel Functions and Random Walks

8. Distributions on a Circle

9. Problems for Solution

Chapter III Densities in Higher Dimensions. Normal Densities and Processes

1. Densities

2. Conditional Distributions

3. Return to the Exponential and the Uniform Distributions

4. A Characterization of the Normal Distribution

5. Matrix Notation. The Covariance Matrix

6. Normal Densities and Distributions

7. Stationary Normal Processes

8. Markovian Normal Densities

9. Problems for Solution

Chapter IV Probability Measures and Spaces

1. Baire Functions

2. Interval Functions and Integrals in R^r

3. σ-Algebras. Measurability

4. Probability Spaces. Random Variables

5. The Extension Theorem

6. Product Spaces. Sequences of Independent Variables

7. Null Sets. Completion

Chapter V Probability Distributions in R^r

1. Distributions and Expectations

2. Preliminaries

3. Densities

4. Convolutions

5. Symmetrization

6. Integration by Parts. Existence of Moments

7. Chebyshev’s Inequality

8. Further Inequalities. Convex Functions

9. Simple Conditional Distributions. Mixtures

10. Conditional Distributions

11. Conditional Expectations

12. Problems for Solution

Chapter VI A Survey of Some Important Distributions and Processes

1. Stable Distributions in R¹

2. Examples

3. Infinitely Divisible Distributions in R¹

4. Processes with Independent Increments

5. Ruin Problems in Compound Poisson Processes

6. Renewal Processes

7. Examples and Problems

8. Random Walks

9. The Queuing Process

10. Persistent and Transient Random Walks

11. General Markov Chains

12. Martingales

13. Problems for Solution

Chapter VII Laws of Large Numbers. Applications in Analysis

1. Main Lemma and Notations

2. Bernstein Polynomials. Absolutely Monotone Functions

3. Moment Problems

4. Application to Exchangeable Variables

5. Generalized Taylor Formula and Semi-Groups

6. Inversion Formulas for Laplace Transforms

7. Laws of Large Numbers for Identically Distributed Variables

8. Strong Laws

9. Generalization to Martingales

10. Problems for Solution

Chapter VIII The Basic Limit Theorems

1. Convergence of Measures

2. Special Properties

3. Distributions as Operators

4. The Central Limit Theorem

5. Infinite Convolutions

6. Selection Theorems

7. Ergodic Theorems for Markov Chains

8. Regular Variation

9. Asymptotic Properties of Regularly Varying Functions

10. Problems for Solution

Chapter IX Infinitely Divisible Distributions and Semi-Groups

1. Orientation

2. Convolution Semi-Groups

3. Preparatory Lemmas

4. Finite Variances

5. The Main Theorems

6. Example: Stable Semi-Groups

7. Triangular Arrays with Identical Distributions

8. Domains of Attraction

9. Variable Distributions. The Three-Series Theorem

10. Problems for Solution

Chapter X Markov Processes and Semi-Groups

1. The Pseudo-Poisson Type

2. A Variant: Linear Increments

3. Jump Processes

4. Diffusion Processes in R¹

5. The Forward Equation. Boundary Conditions

6. Diffusion in Higher Dimensions

7. Subordinated Processes

8. Markov Processes and Semi-Groups

9. The "Exponential Formula" of Semi-Group Theory

10. Generators. The Backward Equation

Chapter XI Renewal Theory

1. The Renewal Theorem

2. Proof of the Renewal Theorem

3. Refinements

4. Persistent Renewal Processes

5. The Number N_t of Renewal Epochs

6. Terminating (Transient) Processes

7. Diverse Applications

8. Existence of Limits in Stochastic Processes

9. Renewal Theory on the Whole Line

10. Problems for Solution

Chapter XII Random Walks in R¹

1. Basic Concepts and Notations

2. Duality. Types of Random Walks

3. Distribution of Ladder Heights. Wiener-Hopf Factorization

3a. The Wiener-Hopf Integral Equation

4. Examples

5. Applications

6. A Combinatorial Lemma

7. Distribution of Ladder Epochs

8. The Arc Sine Laws

9. Miscellaneous Complements

10. Problems for Solution

Chapter XIII Laplace Transforms. Tauberian Theorems. Resolvents

1. Definitions. The Continuity Theorem

2. Elementary Properties

3. Examples

4. Completely Monotone Functions. Inversion Formulas

5. Tauberian Theorems

6. Stable Distributions

7. Infinitely Divisible Distributions

8. Higher Dimensions

9. Laplace Transforms for Semi-Groups

10. The Hille-Yosida Theorem

11. Problems for Solution

Chapter XIV Applications of Laplace Transforms

1. The Renewal Equation: Theory

2. Renewal-Type Equations: Examples

3. Limit Theorems Involving Arc Sine Distributions

4. Busy Periods and Related Branching Processes

5. Diffusion Processes

6. Birth-and-Death Processes and Random Walks

7. The Kolmogorov Differential Equations

8. Example: The Pure Birth Process

9. Calculation of Ergodic Limits and of First-Passage Times

10. Problems for Solution

Chapter XV Characteristic Functions

1. Definition. Basic Properties

2. Special Distributions. Mixtures

2a. Some Unexpected Phenomena

3. Uniqueness. Inversion Formulas

4. Regularity Properties

5. The Central Limit Theorem for Equal Components

6. The Lindeberg Conditions

7. Characteristic Functions in Higher Dimensions

8. Two Characterizations of the Normal Distribution

9. Problems for Solution

Chapter XVI Expansions Related to the Central Limit Theorem,

1. Notations

2. Expansions for Densities

3. Smoothing

4. Expansions for Distributions

5. The Berry-Esséen Theorems

6. Expansions in the Case of Varying Components

7. Large Deviations

Chapter XVII Infinitely Divisible Distributions

1. Infinitely Divisible Distributions

2. Canonical Forms. The Main Limit Theorem

2a. Derivatives of Characteristic Functions

3. Examples and Special Properties

4. Special Properties

5. Stable Distributions and Their Domains of Attraction

6. Stable Densities

7. Triangular Arrays

8. The Class L

9. Partial Attraction. "Universal Laws"

10. Infinite Convolutions

11. Higher Dimensions

12. Problems for Solution 595

Chapter XVIII Applications of Fourier Methods to Random Walks

1. The Basic Identity

2. Finite Intervals. Wald’s Approximation

3. The Wiener-Hopf Factorization

4. Implications and Applications

5. Two Deeper Theorems

6. Criteria for Persistency

7. Problems for Solution

Chapter XIX Harmonic Analysis

1. The Parseval Relation

2. Positive Definite Functions

3. Stationary Processes

4. Fourier Series

5. The Poisson Summation Formula

6. Positive Definite Sequences

7. L² Theory

8. Stochastic Processes and Integrals

9. Problems for Solution

Answers to Problems

Some Books on Cognate Subjects

Index