Probability theory. Edited by K. A. Borovkov. Transl. from the Russian by O. Borovkova and P. S. Ruzankin. (English) Zbl 1297.60002
Universitext. London: Springer (ISBN 978-1-4471-5200-2/pbk; 978-1-4471-5201-9/ebook). xxviii, 733 p. (2013).
This is the English translation of the 5th revised and extended edition of A. A. Borovkov’s book published in 2009 in Russian. For the history of this text, we refer to the reviews of the previous editions [(1999; Zbl 1001.60001), Russian; (1998; Zbl 0918.60003), English; (1986; Zbl 0662.60001), Russian; (1976; Zbl 0345.60001), German; (1976; Zbl 0345.60002), German; (1972; Zbl 0253.60001), Russian].
The current version of the book contains twenty-two chapters and seven appendices. We shortly summarize the content. After a classical introduction to probability theory in the spirit of A. N. Kolmogorov’s “Grundbegriffe” (Chapters 1–8), the author offers a profound and extensive treatment of random processes in discrete time, starting with the classical limit theorems for i.i.d.sequences of random variables and a newly written chapter on their large deviation theory. These are followed by chapters on renewal processes, on zero-one laws for random walks and an updated treatment of their factorization identities. The ergodic theory of Markov chains and the convergence theorems for discrete martingales pave the way for two chapters on the ergodic theory of general stationary sequences and on stochastic recursive sequences. The last five chapters focus on continuous time processes, such as Brownian motion and Lévy processes, and the connection between discrete and continuous time processes via functional limit theorems. The last two chapters introduce general Markov processes in discrete space and Gaussian processes. Three newly added appendices on regularly varying functions, convergence theorems to stable laws as well as on distribution bounds for the sums of i.i.d.sequences round off this monumental work.
Given the amount of the material, the book may well serve as the basis of up to four consecutive, mainly undergraduate probability courses.
Although in the last years many new topics in probability theory have gained a lot of attention, the present author’s account is a precious self-contained standard reference, which preserves and prolongs the excellence of the Soviet probability education to our days.
The current version of the book contains twenty-two chapters and seven appendices. We shortly summarize the content. After a classical introduction to probability theory in the spirit of A. N. Kolmogorov’s “Grundbegriffe” (Chapters 1–8), the author offers a profound and extensive treatment of random processes in discrete time, starting with the classical limit theorems for i.i.d.sequences of random variables and a newly written chapter on their large deviation theory. These are followed by chapters on renewal processes, on zero-one laws for random walks and an updated treatment of their factorization identities. The ergodic theory of Markov chains and the convergence theorems for discrete martingales pave the way for two chapters on the ergodic theory of general stationary sequences and on stochastic recursive sequences. The last five chapters focus on continuous time processes, such as Brownian motion and Lévy processes, and the connection between discrete and continuous time processes via functional limit theorems. The last two chapters introduce general Markov processes in discrete space and Gaussian processes. Three newly added appendices on regularly varying functions, convergence theorems to stable laws as well as on distribution bounds for the sums of i.i.d.sequences round off this monumental work.
Given the amount of the material, the book may well serve as the basis of up to four consecutive, mainly undergraduate probability courses.
Although in the last years many new topics in probability theory have gained a lot of attention, the present author’s account is a precious self-contained standard reference, which preserves and prolongs the excellence of the Soviet probability education to our days.
Reviewer: Michael Högele (Berlin)
MSC:
| 60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |
| 60A05 | Axioms; other general questions in probability |
| 60F05 | Central limit and other weak theorems |
| 60F10 | Large deviations |
