Probability Theory

The aim of this book is to serve as a reference text to provide an orientation in the enormous material which probability theory has accumulated so far. The book mainly treats such topics like the founda­ tions of probability theory, limit theorems and random processes. The bibliography gives a list of the main textbooks on probability theory and its applications. By way of exception some references are planted into the text to recent papers which in our opinion did not find in monographs the attention they deserved (in this connection we do not at all want to attribute any priority to one or the other author). Some references indicate the immediate use of the material taken from the paper in question. In the following we recommend some selected literature, together with indications of the corresponding sections of the present reference book. The textbook by B. V. Gnedenko, "Lehrbuch der Wahrscheinlichkeits­ theorie " , Akademie-Verlag, Berlin 1957, and the book by W. Feller, "IntroductioI). to Probability Theory and its Applications", Wiley, 2. ed., New York 1960 (Chapter I, § 1 of Chapter V) may serve as a first introduction to the various problems of probability theory. A large complex of problems is treated in M. Loeve's monograph "Probability Theory", Van Nostrand, 2. ed., Princeton, N. J.; Toronto, New York, London 1963 (Chapters II, III, § 2 Chapter VI). The foundations of probability theory are given in A. N. Kolmogorov's book "Grund­ begriffe der Wahrscheinlichkeitsrechnung", Springer, Berlin 1933.

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.