عنوان

An introduction to probability theory and its applications /

0471257095

9780471257097

An introduction to probability theory and its applications /

[Book]

William Feller.

Second edition.

New York :

John Wiley & Sons, Inc.,

[1957-1971]

2 volumes :

illustrations ;

24 cm.

Wiley publication in mathematical statistics

Includes bibliographical references and index.

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

0

Introduction to probability theory and its applications.

Probabilities.

Probability.

Probabilités.

Méthodes statistiques.

Probabilidade (Estatistica)

Probabilités.

Probabilités.

Probabilities.

Feller, William,1906-1970.

20200822113951.0

[Book]

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