Probability Theory

I Basic Concepts of Elementary Probability Theory.- § 1. Experiments with Equally Probable Outcomes.- 1. Experiments with a Finite Number of Equally Probable Outcomes.- 2. Some Combinatorial Formulas.- 3. "Geometric" Probabilities.- § 2. The Space of Elementary Events and the Law of Composition of Probabilities.- 1. Combination of Events.- 2. The Space of Elementary Events.- 3. The Law of Composition of Probabilities.- § 3. The Relation between Various Events.- 1. Conditional Probabilities.- 2. Independent Events.- 3. The Amount of Information.- § 4. Random Variables.- 1. Random Variables and their Probability Distributions.- 2. Mathematical Expectation, Variance and Correlation Coefficient.- 3. Integral-Valued Variables and Generating Functions.- § 5. Some Probability Distributions.- 1. Probability Distributions Connected with the Poisson Law.- 2. Probability Distributions Connected with the Normal Law.- 3. Probability Distributions Connected with Bernoulli Trials.- 4. Some Probability Distributions Arising in the Scheme of the Symmetric Random Walk.- II Spaces and Measures.- § 1. Some Facts about Measurable and Topological Spaces.- 1. Measurable and Topological Spaces.- 2. Linear Spaces.- § 2. Distributions and Measures.- 1. Measures in Measurable Spaces.- 2. Measures in Topological Spaces.- 3. Joint Distributions.- § 3. Measures and Integrals.- 1. The Integral and its Properties.- 2. Abstract Measures and Integrals.- III Basic Concepts of Probability Theory.- § 1. Spaces of Elementary Events. Probability Distributions and Characteristic Functions.- 1. Basic Schemes of Probability Theory.- 2. Relations Among Various Events and Random Variables.- 3. Random Processes and their Probability Distributions.- § 2. Basic Types of Random Processes.- 1. Random Processes Considered as Curves in Hilbert Space.- 2. Gaussian Random Processes.- 3. Martingales and Stochastic Integrals.- 4. Markov Random Processes.- 5. Homogeneous and Stationary Random Processes.- IV Limit Theorems in Probability Theory.- § 1. Distributions and their Characteristic Functions.- 1. One-to-one Correspondance between Distributions and Characteristic Functions.- 2. Inversion Formulas.- 3. Properties of Distributions in Terms of Characteristic Functions.- § 2. Estimates of the Nearness of Distributions in Terms of the Nearness of their Characteristic Functions.- 1. Uniform Distances.- 2. The Multi-Dimensional Case.- § 3. Moments and Semi-Invariants.- 1. Formal Relations.- 2. The Moment Problem.- 3. Inequalities.- 4. Convergence of Moments.- § 4. Infinitely Divisible Distributions and their Connection with Limit Theorems.- 1. Definition and Connection with Limit Theorems.- 2. Properties of Infinitely Divisible Laws.- § 5. Sequences of Independent Random Variables (General Properties).- § 6. Sequences of Independent Random Variables. Convergence to the Normal Law.- 1. Conditions for Convergence.- 2. Sharpenings.- 3. The Binomial Distribution.- 4. The Multi-Dimensional Case.- § 7. Sequences of Independent Random Variables. Convergence to Stable Laws.- 1. Definition and Some Properties of Stable Laws.- 2. Conditions for Convergence. Sharpenings.- § 8. Local Theorems for Lattice-Distributions.- 1. Asymptotic Uniform Distributions.- 2. Integral-Valued Identically Distributed Terms.- § 9. Local Theorems for Densities.- § 10. Probabilities of Large Deviations. Inequalities and Asymptotic Formulas.- § 11. Concluding Remarks.- V Markov Processes.- § 1. Markov Processes with a Finite or Denumerable Number of States (Markov Chains).- 1. Markov Property and Transition Probabilities.- 2. Classification of States of a Homogeneous Markov Chain.- 3. Ergodic Properties of Homogeneous Markov Chains.- 4. General Markov Jump Processes.- § 2. Branching Processes.- 1. General Description of a Branching Process.- 2. Branching Processes with One Type of Particles.- § 3. Random Processes with Independent Increments.- 1. Sequences of Sums of an Increasing Number of Independent Random Variables.- 2. The Brownian Motion Process.- 3. Structure of Random Processes with Independent Increments.- § 4. Diffusion Processes.- 1. Differential and Stochastic Equations.- 2. The Behavior of Homogeneous Diffusion Processes on the Boundary Ergodic Properties.- 3. Transformations of Diffusion Processes.- 4. The Kolmogorov Backward Equation and the Probability Distribution of Some Functionals of a Diffusion Process.- 5. Multi-Dimensional Diffusion Processes.- § 5. General Markov Processes and their Characteristics.- 1. Semigroups Corresponding to Transition Functions, and their Infinitesimal Operators.- 2. Infinitesimal Operators, Harmonic and Excessive Functions.- § 6. Controlled Markov Processes.- 1. Controlled Markov Sequences.- 2. Control in the Case of Incomplete Data.- 3. Controlled Diffusion Processes.- VI Stationary Processes.- § 1. Spectral Theory of Harmonizable Processes.- 1. Linear Transformations.- 2. Regular Stationary Processes.- 3. Linear Prediction of Stationary Processes.- 4. Physical Interpretation of the Spectral Representation.- 5. Multi-Dimensional Stationary Processes.- 6. Generalized Stationary Processes and Processes with Stationary Increments.- 7. Harmonizable Random Processes. Some Non-Linear Transformations.- § 2. Stationary Processes in the Strict Sense.- 1. Ergodic Properties.- 2. General Ergodic Properties. Their Application to Markov Processes.- 3. Spectral Conditions for the Ergodicity of Some Stationary Processes.- § 3. Stationary Gaussian Processes.- 1. Some Properties of the Trajectories.- 2. Exits of a Stationary Gaussian Process Beyond a Given Level.- 3. Equivalence of Probability Distributions of Stationary Gaussian Processes.- § 4. Elements of the Mathematical Theory of the Transmission of Information through Stationary Communication Channels.- 1. Fundamental Results on the Possibility of Transmitting Informations.- 2. Formulas for the Amount of Information.