Convergence of probability measures / Patrick Billingsley

Bibliogr. p. 243-247. Notations. Index

This book is a welcome addition to the series of books being written on specialized topics in probability and statistics. The subject matter is of great current interest and the exposition is lucid and elegant.
The author's preface is an accurate summary of this book and hence it suffices to quote from it. The author says: "This book is about weak-convergence methods in metric spaces, with applications sufficient to show their power and utility.
"The introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space of continuous functions on the unit interval and in Chapter 3 to the space of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
"Although standard measure-theoretic probability and metric-space topology are assumed, no general (non-metric) topology is used, and the few results required from functional analysis are proved in the text or in an appendix.
"Mastering the impulse to hoard the examples and applications till the last, thereby obliging the reader to persevere to the end, I have instead spread them evenly through the book to illustrate the theory as it emerges in stages.'' (MathSciNet)