An Introduction to Probability Theory and Its Applications
Subtitle: Volume 2
Edition: 2nd ed.
Series: An Introduction to Probability Theory and Its Applications: Volume 2
Publisher: John Wiley & Sons
Published: 1971
Place of publication: New York, USA
Language: eng
Format: Hardcover
ISBN-13: 9780471257097
Copyright date: 1971
Description: The fundamental character and spirit of this classic remain unchanged, but there are up-dated, revised, and new materials.

This volume features typically thorough coverage--both applied and abstract--of the most "popular" densities, along with the measure-theocratic bases of the theory.

Probabilistic subjects include the laws of large numbers, the central limit theorem, infinitely divisible distributions, Markov processes, random walks and renewal theory. Included, too, in the techniques are Laplace and Fourier transforms, semi-groups and general harmonic analysis.

Experts will find new proofs and results, particularly the rewritten chapter 17. This edition consolidates and unifies the general methodology, obtaining coherence through the resultant simplification.
--back cover
Table of contents: title: Chapter I The Exponential and the Uniform Densities; title: 1. Introduction; title: 2. Densities. Convolutions; title: 3. The Exponential Density; title: 4. Waiting Time Paradoxes. The Poisson Process; title: 5. The Persistence of Bad Luck; title: 6. Waiting Times and Order Statistics; title: 7. The Uniform Distribution; title: 8. Random Splittings; title: 9. Convolutions and Covering Theorems; title: 10. Random Directions; title: 11. The Use of Lebesgue Measure; title: 12. Empirical Distributions; title: 13. Problems for Solution; title: Chapter II Special Densities. Randomization; title: 1. Notations and Conventions; title: 2. Gamma Distributions; title: 3. Related Distributions of Statistics; title: 4. Some Common Densities; title: 5. Randomization and Mixtures; title: 6. Discrete Distributions; title: 7. Bessel Functions and Random Walks; title: 8. Distributions on a Circle; title: 9. Problems for Solution; title: Chapter III Densities in Higher Dimensions. Normal Densities and Processes; title: 1. Densities; title: 2. Conditional Distributions; title: 3. Return to the Exponential and the Uniform Distributions; title: 4. A Characterization of the Normal Distribution; title: 5. Matrix Notation. The Covariance Matrix; title: 6. Normal Densities and Distributions; title: 7. Stationary Normal Processes; title: 8. Markovian Normal Densities; title: 9. Problems for Solution; title: Chapter IV Probability Measures and Spaces; title: 1. Baire Functions; title: 2. Interval Functions and Integrals in Rr; title: 3. σ-Algebras. Measurability; title: 4. Probability Spaces. Random Variables; title: 5. The Extension Theorem; title: 6. Product Spaces. Sequences of Independent Variables; title: 7. Null Sets. Completion; title: Chapter V Probability Distributions in Rr; title: 1. Distributions and Expectations; title: 2. Preliminaries; title: 3. Densities; title: 4. Convolutions; title: 5. Symmetrization; title: 6. Integration by Parts. Existence of Moments; title: 7. Chebyshev’s Inequality; title: 8. Further Inequalities. Convex Functions; title: 9. Simple Conditional Distributions. Mixtures; title: 10. Conditional Distributions; title: 11. Conditional Expectations; title: 12. Problems for Solution; title: Chapter VI A Survey of Some Important Distributions and Processes; title: 1. Stable Distributions in R1; title: 2. Examples; title: 3. Infinitely Divisible Distributions in R1; title: 4. Processes with Independent Increments; title: 5. Ruin Problems in Compound Poisson Processes; title: 6. Renewal Processes; title: 7. Examples and Problems; title: 8. Random Walks; title: 9. The Queuing Process; title: 10. Persistent and Transient Random Walks; title: 11. General Markov Chains; title: 12. Martingales; title: 13. Problems for Solution; title: Chapter VII Laws of Large Numbers. Applications in Analysis; title: 1. Main Lemma and Notations; title: 2. Bernstein Polynomials. Absolutely Monotone Functions; title: 3. Moment Problems; title: 4. Application to Exchangeable Variables; title: 5. Generalized Taylor Formula and Semi-Groups; title: 6. Inversion Formulas for Laplace Transforms; title: 7. Laws of Large Numbers for Identically Distributed Variables; title: 8. Strong Laws; title: 9. Generalization to Martingales; title: 10. Problems for Solution; title: Chapter VIII The Basic Limit Theorems; title: 1. Convergence of Measures; title: 2. Special Properties; title: 3. Distributions as Operators; title: 4. The Central Limit Theorem; title: 5. Infinite Convolutions; title: 6. Selection Theorems; title: 7. Ergodic Theorems for Markov Chains; title: 8. Regular Variation; title: 9. Asymptotic Properties of Regularly Varying Functions; title: 10. Problems for Solution; title: Chapter IX Infinitely Divisible Distributions and Semi-Groups; title: 1. Orientation; title: 2. Convolution Semi-Groups; title: 3. Preparatory Lemmas; title: 4. Finite Variances; title: 5. The Main Theorems; title: 6. Example: Stable Semi-Groups; title: 7. Triangular Arrays with Identical Distributions; title: 8. Domains of Attraction; title: 9. Variable Distributions. The Three-Series Theorem; title: 10. Problems for Solution; title: Chapter X Markov Processes and Semi-Groups; title: 1. The Pseudo-Poisson Type; title: 2. A Variant: Linear Increments; title: 3. Jump Processes; title: 4. Diffusion Processes in R1; title: 5. The Forward Equation. Boundary Conditions; title: 6. Diffusion in Higher Dimensions; title: 7. Subordinated Processes; title: 8. Markov Processes and Semi-Groups; title: 9. The "Exponential Formula" of Semi-Group Theory; title: 10. Generators. The Backward Equation; title: Chapter XI Renewal Theory; title: 1. The Renewal Theorem; title: 2. Proof of the Renewal Theorem; title: 3. Refinements; title: 4. Persistent Renewal Processes; title: 5. The Number Nt of Renewal Epochs; title: 6. Terminating (Transient) Processes; title: 7. Diverse Applications; title: 8. Existence of Limits in Stochastic Processes; title: 9. Renewal Theory on the Whole Line; title: 10. Problems for Solution; title: Chapter XII Random Walks in R1; title: 1. Basic Concepts and Notations; title: 2. Duality. Types of Random Walks; title: 3. Distribution of Ladder Heights. Wiener-Hopf Factorization; title: 3a. The Wiener-Hopf Integral Equation; title: 4. Examples; title: 5. Applications; title: 6. A Combinatorial Lemma; title: 7. Distribution of Ladder Epochs; title: 8. The Arc Sine Laws; title: 9. Miscellaneous Complements; title: 10. Problems for Solution; title: Chapter XIII Laplace Transforms. Tauberian Theorems. Resolvents; title: 1. Definitions. The Continuity Theorem; title: 2. Elementary Properties; title: 3. Examples; title: 4. Completely Monotone Functions. Inversion Formulas; title: 5. Tauberian Theorems; title: 6. Stable Distributions; title: 7. Infinitely Divisible Distributions; title: 8. Higher Dimensions; title: 9. Laplace Transforms for Semi-Groups; title: 10. The Hille-Yosida Theorem; title: 11. Problems for Solution; title: Chapter XIV Applications of Laplace Transforms; title: 1. The Renewal Equation: Theory; title: 2. Renewal-Type Equations: Examples; title: 3. Limit Theorems Involving Arc Sine Distributions; title: 4. Busy Periods and Related Branching Processes; title: 5. Diffusion Processes; title: 6. Birth-and-Death Processes and Random Walks; title: 7. The Kolmogorov Differential Equations; title: 8. Example: The Pure Birth Process; title: 9. Calculation of Ergodic Limits and of First-Passage Times; title: 10. Problems for Solution; title: Chapter XV Characteristic Functions; title: 1. Definition. Basic Properties; title: 2. Special Distributions. Mixtures; title: 2a. Some Unexpected Phenomena; title: 3. Uniqueness. Inversion Formulas; title: 4. Regularity Properties; title: 5. The Central Limit Theorem for Equal Components; title: 6. The Lindeberg Conditions; title: 7. Characteristic Functions in Higher Dimensions; title: 8. Two Characterizations of the Normal Distribution; title: 9. Problems for Solution; title: Chapter XVI Expansions Related to the Central Limit Theorem,; title: 1. Notations; title: 2. Expansions for Densities; title: 3. Smoothing; title: 4. Expansions for Distributions; title: 5. The Berry-Esséen Theorems; title: 6. Expansions in the Case of Varying Components; title: 7. Large Deviations; title: Chapter XVII Infinitely Divisible Distributions; title: 1. Infinitely Divisible Distributions; title: 2. Canonical Forms. The Main Limit Theorem; title: 2a. Derivatives of Characteristic Functions; title: 3. Examples and Special Properties; title: 4. Special Properties; title: 5. Stable Distributions and Their Domains of Attraction; title: 6. Stable Densities; title: 7. Triangular Arrays; title: 8. The Class L; title: 9. Partial Attraction. "Universal Laws"; title: 10. Infinite Convolutions; title: 11. Higher Dimensions; title: 12. Problems for Solution 595; title: Chapter XVIII Applications of Fourier Methods to Random Walks; title: 1. The Basic Identity; title: 2. Finite Intervals. Wald’s Approximation; title: 3. The Wiener-Hopf Factorization; title: 4. Implications and Applications; title: 5. Two Deeper Theorems; title: 6. Criteria for Persistency; title: 7. Problems for Solution; title: Chapter XIX Harmonic Analysis; title: 1. The Parseval Relation; title: 2. Positive Definite Functions; title: 3. Stationary Processes; title: 4. Fourier Series; title: 5. The Poisson Summation Formula; title: 6. Positive Definite Sequences; title: 7. L2 Theory; title: 8. Stochastic Processes and Integrals; title: 9. Problems for Solution; title: Answers to Problems; title: Some Books on Cognate Subjects; title: Index