An Introduction to Probability Theory and Its Applications, Volume 2

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 Rr 3. sigma-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 Rr 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 Nt 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-Esseen 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