Albert N. Shiryaev - Probability-2

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

More information about this series at http://www.springer.com/series/136

This Springer imprint is published by the registered company Springer Science+Business Media, LLC part of Springer Nature.

The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A.

The present edition is a translation of the fourth Russian edition of 2007, with the previous three published in 1980, 1989, and 2004. The English translations of the first two appeared in 1984 and 1996. The third and fourth Russian editions, extended compared to the second edition, were published in two volumes titled Probability-1 and Probability-2 . Accordingly, the present edition consists of two volumes: this Vol. 2, titled Probability-2 , contains Chaps. , and Chaps. 13 are contained in Vol. 1, titled Probability-1 , which was published in 2016.

This English translation of Probability-2 was prepared by the editor and translator Prof. D. M. Chibisov, Professor of the Steklov Mathematical Institute. A former student of N. V. Smirnov, he has a broad view of probability and mathematical statistics, which enabled him not only to translate the parts that had not been translated before, but also to edit both the previous translation and the Russian text, making in them quite a number of corrections and amendments.

The author is sincerely grateful to D. M. Chibisov for the translation and scientific editing of this book.

A university course on probability and statistics usually consists of three one-semester parts: probability theory, random processes, and mathematical statistics.

The book Probability-1 covered the material normally included in probability theory.

This book, Probability-2 , contains extensive material for a course on random processes in the part dealing with discrete time processes, i.e., random sequences. (The reader interested in random processes with continuous time may refer to [12], which is closely related to Probability-1 and Probability-2 .)

Chapter , which opens this book, is focused mostly on the properties of sums of independent random variables that hold with probability one (e.g., zeroone laws, the strong law of large numbers, the law of the iterated logarithm).

Chapters treat the strict and wide sense stationary random sequences.

In Chaps. and , we set out random sequences that form martingales and Markov chains . These classes of processes enable us to study the behavior of various stochastic systems in the future, depending on their past and present thanks to which these processes play a very important role in modern probability theory and its applications.

The book concludes with a Historical Review of the Development of Mathematical Theory of Probability.