(This review is based on the second Russian edition printed in 1986. The contents of English edition seems to be a bit extended version of the Russian one.) This is a very good exposition to probability theory at the professional level. I like it much more than I do Billingsley's "Probability and Measure" (which is a collection of essays on probability theory, sometimes only vaguely related a few chapters apart from each other, while Borovkov is a very consistent course which has the same level of rigor as Billingsley does, and which also proves some subtle but appreciable things not mentioned in Billingsley). It has a bit different flavor of tending to prove things via characteristic functions rather than directly with the cesnored random variables as in Billinsley's book. It does not cover as much measure theory as Billingsley does, and I suspect that the book implicitly assumes the student to be familiar with a standard Russian reference on functional analysis by Kolmogorov and Fomin that has an extensive treatment of Lebesgue integration. It is also nice that it has a lot of examples discussed in the text (rather than given as exercises) that help to cement the concepts. They make the text quite lively, too. Sometimes I had to spend a minute or two thinking why they believe a statement in a proof is self-evident, though. The book starts with with the introduction of probability spaces, goes on to random variables, then to the laws of large numbers, convergence notions, and the CLTs. It also discusses renewal theory, factorization identities, Markov chains, information and entropy, martingales, continuous time stocastic processes, functional limit theorems, and Markov processes, with some measure theory stuff and a couple of more difficult theorems (extension of a measure, Kolmogorov theorem on consistent distributions, theorems of Helly and Arcela--Ascoli) given in appendices. Thus it covers more than a semester of probability theory, giving some initial reading for some four or so advanced courses. The author suggests to use the bulk of the material in the first ten or twelve chapters for a required semester course, with the rest of the book viewed as the material for shorter elective courses. The book helped me greatly in my probability theory comprehensive exam, as well as in my stochastic calculus and stochastic processes courses. It is a pity that the book is rather expensive -- I am happy to have it in the original language on which this review is actually based. Of historical interest it is that A.Borovkov is Kolmogorov's student. |