Introduction to Probability Theory and Its Applications, Volume 2

William "Vilim" Feller was a Croatian-American mathematician specializing in probability theory.

• Chapter I The Exponential and the Uniform Densities 1. Introduction2. Densities. Convolutions3. The Exponential Density4. Waiting Time Paradoxes. The Poisson Process5. The Persistence of Bad Luck6. Waiting Times and Order Statistics7. The Uniform Distribution8. Random Splittings9. Convolutions and Covering Theorems10. Random Directions11. The Use of Lebesgue Measure12. Empirical Distributions13. Problems for SolutionChapter II Special Densities. Randomization1. Notations and Conventions2. Gamma Distributions3. Related Distributions of Statistics4. Some Common Densities5. Randomization and Mixtures6. Discrete Distributions7. Bessel Functions and Random Walks8. Distributions on a Circle9. Problems for SolutionChapter III Densities in Higher Dimensions. Normal Densities and Processes1. Densities2. Conditional Distributions3. Return to the Exponential and the Uniform Distributions4. A Characterization of the Normal Distribution5. Matrix Notation. The Covariance Matrix6. Normal Densities and Distributions7. Stationary Normal Processes8. Markovian Normal Densities9. Problems for SolutionChapter IV Probability Measures and Spaces1. Baire Functions2. Interval Functions and Integrals in Rr3. σ-Algebras. Measurability4. Probability Spaces. Random Variables5. The Extension Theorem6. Product Spaces. Sequences of Independent Variables7. Null Sets. CompletionChapter V Probability Distributions in Rr1. Distributions and Expectations2. Preliminaries3. Densities4. Convolutions5. Symmetrization6. Integration by Parts. Existence of Moments7. Chebyshev’s Inequality8. Further Inequalities. Convex Functions9. Simple Conditional Distributions. Mixtures10. Conditional Distributions11. Conditional Expectations12. Problems for SolutionChapter VI A Survey of Some Important Distributions and Processes1. Stable Distributions in R12. Examples3. Infinitely Divisible Distributions in R14. Processes with Independent Increments5. Ruin Problems in Compound Poisson Processes6. Renewal Processes7. Examples and Problems8. Random Walks9. The Queuing Process10. Persistent and Transient Random Walks11. General Markov Chains12. Martingales13. Problems for SolutionChapter VII Laws of Large Numbers. Applications in Analysis1. Main Lemma and Notations2. Bernstein Polynomials. Absolutely Monotone Functions3. Moment Problems4. Application to Exchangeable Variables5. Generalized Taylor Formula and Semi-Groups6. Inversion Formulas for Laplace Transforms7. Laws of Large Numbers for Identically Distributed Variables8. Strong Laws9. Generalization to Martingales10. Problems for SolutionChapter VIII The Basic Limit Theorems1. Convergence of Measures2. Special Properties3. Distributions as Operators4. The Central Limit Theorem5. Infinite Convolutions6. Selection Theorems7. Ergodic Theorems for Markov Chains8. Regular Variation9. Asymptotic Properties of Regularly Varying Functions10. Problems for SolutionChapter IX Infinitely Divisible Distributions and Semi-Groups1. Orientation2. Convolution Semi-Groups3. Preparatory Lemmas4. Finite Variances5. The Main Theorems6. Example: Stable Semi-Groups7. Triangular Arrays with Identical Distributions8. Domains of Attraction9. Variable Distributions. The Three-Series Theorem10. Problems for SolutionChapter X Markov Processes and Semi-Groups1. The Pseudo-Poisson Type2. A Variant: Linear Increments3. Jump Processes4. Diffusion Processes in R15. The Forward Equation. Boundary Conditions6. Diffusion in Higher Dimensions7. Subordinated Processes8. Markov Processes and Semi-Groups9. The "Exponential Formula" of Semi-Group Theory10. Generators. The Backward EquationChapter XI Renewal Theory1. The Renewal Theorem2. Proof of the Renewal Theorem3. Refinements4. Persistent Renewal Processes5. The Number Nt of Renewal Epochs6. Terminating (Transient) Processes7. Diverse Applications8. Existence of Limits in Stochastic Processes9. Renewal Theory on the Whole Line10. Problems for SolutionChapter XII Random Walks in R11. Basic Concepts and Notations2. Duality. Types of Random Walks3. Distribution of Ladder Heights. Wiener-Hopf Factorization3a. The Wiener-Hopf Integral Equation4. Examples5. Applications6. A Combinatorial Lemma7. Distribution of Ladder Epochs8. The Arc Sine Laws9. Miscellaneous Complements10. Problems for SolutionChapter XIII Laplace Transforms. Tauberian Theorems. Resolvents1. Definitions. The Continuity Theorem2. Elementary Properties3. Examples4. Completely Monotone Functions. Inversion Formulas5. Tauberian Theorems6. Stable Distributions7. Infinitely Divisible Distributions8. Higher Dimensions9. Laplace Transforms for Semi-Groups10. The Hille-Yosida Theorem11. Problems for SolutionChapter XIV Applications of Laplace Transforms1. The Renewal Equation: Theory2. Renewal-Type Equations: Examples3. Limit Theorems Involving Arc Sine Distributions4. Busy Periods and Related Branching Processes5. Diffusion Processes6. Birth-and-Death Processes and Random Walks7. The Kolmogorov Differential Equations8. Example: The Pure Birth Process9. Calculation of Ergodic Limits and of First-Passage Times10. Problems for SolutionChapter XV Characteristic Functions1. Definition. Basic Properties2. Special Distributions. Mixtures2a. Some Unexpected Phenomena3. Uniqueness. Inversion Formulas4. Regularity Properties5. The Central Limit Theorem for Equal Components6. The Lindeberg Conditions7. Characteristic Functions in Higher Dimensions8. Two Characterizations of the Normal Distribution9. Problems for SolutionChapter XVI Expansions Related to the Central Limit Theorem,1. Notations2. Expansions for Densities3. Smoothing4. Expansions for Distributions5. The Berry-Esséen Theorems6. Expansions in the Case of Varying Components7. Large DeviationsChapter XVII Infinitely Divisible Distributions1. Infinitely Divisible Distributions2. Canonical Forms. The Main Limit Theorem2a. Derivatives of Characteristic Functions3. Examples and Special Properties4. Special Properties5. Stable Distributions and Their Domains of Attraction6. Stable Densities7. Triangular Arrays8. The Class L9. Partial Attraction. "Universal Laws"10. Infinite Convolutions11. Higher Dimensions12. Problems for Solution 595Chapter XVIII Applications of Fourier Methods to Random Walks1. The Basic Identity2. Finite Intervals. Wald’s Approximation3. The Wiener-Hopf Factorization4. Implications and Applications5. Two Deeper Theorems6. Criteria for Persistency7. Problems for SolutionChapter XIX Harmonic Analysis1. The Parseval Relation2. Positive Definite Functions3. Stationary Processes4. Fourier Series5. The Poisson Summation Formula6. Positive Definite Sequences7. L2 Theory8. Stochastic Processes and Integrals9. Problems for SolutionAnswers to ProblemsSome Books on Cognate SubjectsIndex