An Introduction to Probability Theory and Its Applications, Vol 2, 2ed, An Indian Adaptation
Description
An Introduction to Probability Theory and Its Applications offers comprehensive explanations to complex statistical problems. Delving deep into densities and distributions while relating critical formulas, processes, and approaches, this rigorous text provides a solid grounding in probability with practice problems throughout. Emphasizing application without neglecting theory, the text carefully explains complex topics and their practical use.
This Indian edition enhances the coverage with new and updated content. It introduces new sections discussing nonnormal densities in higher dimensions, statement of Hájek−Rényi inequality, and Kolmogorov's strong law of large numbers. Additionally, a new appendix , featuring simplified real-world examples using R software, has been added. Many end-of-chapter problems throughout the book are new or revised. It also includes multiple-choice questions derived from various competitive examinations.
Table of Contents
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. Summary 14. Problems for Solution 15. Multiple Choice Questions 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. Summary 10. Problems for Solution 11. Multiple Choice Questions 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. Non – Normal Densities in Higher Dimensions 10. Summary 11. Problems for Solution 12. Multiple Choice Questions IV Probability Measures and Spaces 1. Baire Functions 2. Interval Functions and Integrals in ℛr 3. σ-Algebras. Measurability 4. Probability Spaces. Random Variables 5. The Extension Theorem 6. Product Spaces. Sequences of Independent Variables 7. Null Sets. Completion 8. Summary 9. Multiple Choice Questions 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. Summary 13. Problems for Solution 14. Multiple Choice Questions 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. Summary 14. Problems for Solution 15. Multiple Choice Questions 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. Statement of Hájek−Rényi Inequality 8. Laws of Large Numbers for Identically Distributed Variables 9. Strong Laws 10. Kolmogorov’s Strong Law of Large Numbers 11. Generalization to Martingales 12. Summary 13. Problems for Solution 14. Multiple Choice Questions 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. Summary 11. Problems for Solution 12. Multiple Choice Questions 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. Summary 11. Problems for Solution 12. Multiple Choice Questions 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 11. Summary 12. Problems for Solution 13. Multiple Choice Questions 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. Summary 11. Problems for Solution 12. Multiple Choice Questions XII Random Walks in ℛ1 1. Basic Concepts and Notations 2. Duality. Types of Random Walks 3. Distribution of Ladder Heights. Wiener-Hopf Factorization 4. Examples 5. Applications 6. A Combinatorial Lemma 7. Distribution of Ladder Epochs 8. The Arc Sine Laws 9. Miscellaneous Complements 10. Summary 11. Problems for Solution 12. Multiple Choice Questions 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. Summary 12. Problems for Solution 13. Multiple Choice Questions 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. Summary 11. Problems for Solution 12. Multiple Choice Questions
XV Characteristic Functions 1. Definition. Basic Properties 2. Special Distributions. Mixtures 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. Summary 10. Problems for Solution 11. Multiple Choice Questions 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 8. Summary 9. Problems for Solution 10. Multiple Choice Questions XVII Infinitely Divisible Distributions 1. Infinitely Divisible Distributions 2. Canonical Forms. The Main Limit Theorem 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. Summary 13. Problems for Solution 14. Multiple Choice Questions 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. Summary 8. Problems for Solution 9. Multiple Choice Questions 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. Summary 10. Problems for Solution 11. Multiple Choice Questions Appendix: Examples with R Answers to Problems Some Books on Cognate Subjects Index |