Convergence of probability measures. 2nd ed. (English) Zbl 0944.60003

Billingsley, Patrick

Wiley Series in Probability and Statistics. Chichester: Wiley. x, 277 p. (1999).

It seems safe to assume that most probabilists are familiar with the first edition of the author’s classical book on “Weak convergence of probability measures” (1968; Zbl 0172.21201). This second edition is more than just a reprinted version. It takes up the classical theory, but presents it in an often simplified and streamlined manner and adds a number of topics that have been developed since the appearance of the first edition more than 30 years ago.
The book consists of five chapters. Chapters 1 (Weak convergence in metric spaces), Chapter 2 (The space \(C\)) and Chapter 3 (The space \(D\)) are fairly similar to the first edition although the order of topics has been changed in some places and the overall presentation has been tightened. The material on separable stochastic processes and the entire old Section 7 (Characteristic functions, local limit theorems, some examples) have been removed. Prokhorov’s theorem is proved by a different method, empirical processes are now only treated within the setting of the space \(D\) and the maximal inequalities for dependent random variables in Chapter 3 are proved much more efficiently. In addition, there are some new sections. In Chapter 1, Section 4 treats the asymptotics of long cycles of a random permutation and of large prime divisors of a random integer, and Section 6 contains miscellaneous useful results, including Skorokhod’s representation theorem and the Prokhorov metric. Chapter 2 presents in Section 11 functional limit theorems for lacunary series, and Chapter 3 has new Sections 15 on uniform topologies with applications to empirical processes and 16 on the space \(D[0,\infty)\). The Poisson limit case is now also briefly treated in Chapter 3. Chapter 4 (Dependent variables) has changed quite drastically. By using Gordin’s method of approximating stationary sequences by martingales, the number of pages here has been cut by more than half even after inclusion of a new second section on prime divisors of a random integer. The final Chapter 5 (Other modes of convergence) is entirely new. It treats Strassen’s theorem and a version of Donsker’s theorem with convergence in probability instead of weak convergence.
At $ 80 or £65, the book is not low-priced. But it seems destined to become another classic and is of interest even to those who already own the first edition.

Reviewer: M.Schweizer (Berlin)

MSC:

60-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60B10	Convergence of probability measures
28A33	Spaces of measures, convergence of measures
60B05	Probability measures on topological spaces

Citations:

Zbl 0172.21201