Probability

Preface to the Second Edition Preface to the First Edition Introduction CHAPTER ¢ñ Elementary Probability Theory ¡ì1.Probabilistic Model of an Experiment with a Finite Number of Outcomes ¡ì2.Some Classical Models and Distributions ¡ì3.Conditional Probability.Independence ¡ì4.Random Variables and Their Properties ¡ì5.The Bernoulli Scheme. ¢ñ. The Law of Large Numbers ¡ì6.The Bernoulli Scheme. ¢ò. Limit Theorems (Local, De Moivre-Laplace, Poisson) ¡ì7.Estimating the Probability of Success in the Bernoulli Scheme ¡ì8.Conditional Probabilities and Mathematical Expectations with Respect to Decompositions ¡ì9.Random Walk. ¢ñ. Probabilities of Ruin and Mean Duration in Coin Tossing ¡ì10.Random Walk. ¢ò. Reflection Principle.Arcsine Law ¡ì11.Martingales. Some Applications to the Random Walk ¡ì12.Markov Chains. Ergodic Theorem. Strong Markov Property CHAPTER ¢ò Mathematical Foundations of Probability Theory ¡ì1.Probabilistic Model for an Experiment with Infinitely Many Outcomes. Kolmogorov's Axioms ¡ì2.Algebras and o-algebras. Measurable Spaces ¡ì3.Methods of Introducing Probability Measures on Measurable Spaces ¡ì4.Random Variables. ¢ñ. ¡ì5.Random Elements ¡ì6.Lebesgue Integral.Expectation ¡ì7.Conditional Probabilities and Conditional Expectations with Respect to a o-Algebra ¡ì8.Random Variables. ¢ò. ¡ì9.Construction of a Process with Given Finite-Dimensional Distribution ¡ì10.Various Kinds of Convergence of Sequences of Random Variables ¡ì11.The Hilbert Space of Random Variables with Finite Second Moment ¡ì12.Characteristic Functions ¡ì13.Gaussian Systems CHAPTER ¢ó Convergence of Probability Measures.Central Limit Theorem ¡ì1.Weak Convergence of Probability Measures and Distributions ¡ì2.Relative Compactness and Tightness of Families of Probability Distributions ¡ì3.Proofs of Limit Theorems by the Method of Characteristic Functions ¡ì4.Central Limit Theorem for Sums of Independent Random Variables. ¢ñ. The Lindeberg Condition ¡ì5.Central Limit Theorem for Sums of Independent Random Variables. ¢ò. Nonclassical Conditions ¡ì6.Infinitely Divisible and Stable Distributions ¡ì7.Metrizability of Weak Convergence ¡ì8.On the Connection of Weak Convergence of Measures with Almost Sure Convergence of Random Elements ("Method of a Single Probability Space") ¡ì9.The Distance in Variation between Probability Measures. Kakutani-Hellinger Distance and Hellinger Integrals. Application to Absolute Continuity and Singularity of Measures ¡ì10.Contiguity and Entire Asymptotic Separation of Probability Measures ¡ì11.Rapidity of Convergence in the Central Limit Theorem ¡ì12.Rapidity of Convergence in Poisson's Theorem CHAPTER ¢ô Sequences and Sums of Independent Random Variables ¡ì1.Zero-or-One Laws ¡ì2.Convergence of Series ¡ì3.Strong Law of Large Numbers ¡ì4.Law of the Iterated Logarithm ¡ì5.Rapidity of Convergence in the Strong Law of Large Numbers and in the Probabilities of Large Deviations CHAPTER ¢õ Stationary (Strict Sense) Random Sequences and Ergodic Theory ¡ì1.Stationary (Strict Sense) Random Sequences.Measure-Preserving Transformations ¡ì2.Ergodicity and Mixing ¡ì3.Ergodic Theorems CHAPTER ¢ö Stationary (Wide Sense) Random Sequences. L2 Theory ¡ì1.Spectral Representation of the Covariance Function ¡ì2.Orthogonal Stochastic Measures and Stochastic Integrals ¡ì3.Spectral Representation of Stationary (Wide Sense) Sequences ¡ì4.Statistical Estimation of the Covariance Function and the Spectral Density ¡ì5.Wold's Expansion ¡ì6.Extrapolation, Interpolation and Filtering ¡ì7.The Kalman-Bucy Filter and Its Generalizations CHAPTER ¢÷ Sequences of Random Variables that Form Martingales ¡ì1.Definitions of Martingales and Related Concepts ¡ì2.Preservation of the Martingale Property Under Time Change at a Random Time ¡ì3.Fundamental Inequalities ¡ì4.General Theorems on the Convergence of Submartingales and Martingales ¡ì5.Sets of Convergence of Submartingales and Martingales ¡ì6.Absolute Continuity and Singularity of Probability Distributions ¡ì7.Asymptotics of the Probability of the Outcome of a Random Walk with Curvilinear Boundary ¡ì8.Central Limit Theorem for Sums of Dependent Random Variables ¡ì9.Discrete Version of Ito's Formula ¡ì10.Applications to Calculations of the Probability of Ruin in Insurance CHAPTER ¢ø Sequences of Random Variables that Form Markov Chains ¡ì1.Definitions and Basic Properties ¡ì2.Classification of the States of a Markov Chain in Terms of Arithmetic Properties of the Transition Probabilities p(n)ij ¡ì3.Classification of the States of a Markov Chain in Terms of Asymptotic Properties of the Probabilities p(n)ii ¡ì4.On the Existence of Limits and of Stationary Distributions ¡ì5.Examples Historical and Bibliographical Notes References Index of Symbols Index