Vaart, Aad van der; Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. [B] Springer Series in Statistics. New York, NY: Springer, Berlin, 1996. 508 pp., 34.50 English Pounds. [ISBN 0-387-94640-3]
This book tries to achieve three goals and is consequently divided into three parts.
The first part (`Stochastic convergence') is a systematic exposition of modern weak convergence theory, which generalizes the classical theory of weak convergence to non-measurable maps. This theory, initiated by J. Hoffmann-Joergensen, still allows us to recover many results of classical theory (as treated in Billingsley (1968)). After providing the basic elements of the theory the relationship to classical theory is explored. Then the appropriate analogues of the notions of `convergence in probability' and `almost sure convergence' are developed and investigated. This part provides the tools with which empirical processes and asymptotic statistical theory are studied in the subsequent parts.
The second part (`Empirical processes') presents an account of the major components of the modern theory of empirical processes indexed by classes of sets and functions. The main results in this part concern the uniform law of large numbers (Glivenko-Cantelli theorem) and the uniform central limit theorem (Donsker theorem).
The third part (`Statistical applications') uses modern weak convergence theory and modern empirical process theory to discuss a wide range of applications in statistics. This part includes rates of convergence in semiparametric estimation, the functional delta method, the bootstrap, permutation empirical processes and the convolution theorem.
I think that the authors have achieved their goals to a great extent. This book not only gives easy access to the striking recent developments in weak convergence theory and empirical process theory but also provides, through its well-chosen set of applications (and problems and complements parts), splendid stimulation for further study and research. It succeeds and complements Billingsley's classic work and will become the standard source of study and reference for students and researchers in the area of weak convergence and empirical process theory.
References
Billingsley, P. (1968) Convergence of Probability Measures. New York, Wiley.
Ruediger Kiesel
Birkbeck College
London