An Introduction to Probability Theory and Its Applications, Volume 2

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

Klappentext
The classic text for understanding complex statistical probability 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. Heavy on application without sacrificing theory, the discussion takes the time to explain difficult topics and how to use them. This new second edition includes new material related to the substitution of probabilistic arguments for combinatorial artifices as well as new sections on branching processes, Markov chains, and the DeMoivre-Laplace theorem.

Zusammenfassung
The classic text for understanding complex statistical probability An Introduction to Probability Theory and Its Applications offers comprehensive explanations to complex statistical problems.

Inhalt
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 R r. 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 R r. 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 R 1. 2. Examples. 3. Infinitely Divisible Distributions in R 1. 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 R 1. 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 N t of Renewal Epochs. 6. Terminating (Transient) Processes. 7. Diverse Applications. 8. E

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