Discrete Spaces of Elementary Events.- An Arbitrary Space of Elementary Events.- Random Variables and Distribution Functions.- Numerical Characteristics of Random Variables.- Sequences of Independent Trials with Two Outcomes.- On Convergence of Random Variables and Distributions.- Characteristic Functions.- Sequences of Independent Random Variables. Limit Theorems.- Large Deviation Probabilities for Sums of Independent Random Variables.- Renewal Processes.- Properties of the Trajectories of Random Walks. Zero-One Laws.- Random Walks and Factorisation Identities.- Sequences of Dependent Trials. Markov Chains.- Information and Entropy.- Martingales.- Stationary Sequences.- Stochastic Recursive Sequences.- Continuous Time Random Processes.- Processes with Independent Increments.- Functional Limit Theorems.- Markov Processes.- Processes with Finite Second Moments. Gaussian Processes.- Appendices.
Probability theory is an actively developing branch of mathematics. It has applications in many areas of science and technology and forms the basis of mathematical statistics. This self-contained, comprehensive book tackles the principal problems and advanced questions of probability theory and random processes in 22 chapters, presented in a logical order but also suitable for dipping into. They include both classical and more recent results, such as large deviations theory, factorization identities, information theory, stochastic recursive sequences. The book is further distinguished by the inclusion of clear and illustrative proofs of the fundamental results that comprise many methodological improvements aimed at simplifying the arguments and making them more transparent.
The importance of the Russian school in the development of probability theory has long been recognized. This book is the translation of the fifth edition of the highly successful and esteemed Russian textbook. This edition includes a number of new sections, such as a new chapter on large deviation theory for random walks, which are of both theoretical and applied interest. The frequent references to Russian literature throughout this work lend a fresh dimension and makes it an invaluable source of reference for Western researchers and advanced students in probability related subjects.
Probability Theory will be of interest to both advanced undergraduate and graduate students studying probability theory and its applications. It can serve as a basis for several one-semester courses on probability theory and random processes as well as self-study.
About the Author
Professor Alexandr Borovkov lives and works in the Novosibirsk Academy Town in Russia and is affiliated with both
Presents a wide range of results in logic and computational complexity
Explains the topic informally and then in more detail for the advanced reader
Presents the ideas behind the theoretical concepts