Convergence of stochastic processes. (English) Zbl 0544.60045

Pollard, David

Springer Series in Statistics. New York etc.: Springer-Verlag. XIV, 215 p. DM 82.00 (1984).

This is a clearly written book on certain aspects of asymptotic stochastic process theory. Special emphasis is given to measure theoretic, combinatorial and path orientated problems in general empirical process theory as initiated by R. M. Dudley. Chapters are as following: Functionals on stochastic processes; Uniform convergence of empirical measures; Convergence in distribution in Euclidean spaces; Convergence in distribution in metric spaces; The uniform metric on spaces of cadlag functions; The Skorohod metric on D[0,\(\infty)\); Central limit theorems; Martingales.
There is only few overlap with P. Billingsley’s treatise on weak convergence [Weak convergence of measures: Applications in probability. (1971; Zbl 0271.60009)]. Some of the topics which seem worthwhile further mentioning are: \(\sigma\)-fields generated by balls, weak convergence on non Borel \(\sigma\)-fields, coupling, chaining, covering numbers, clustering.

Reviewer: W.Stute

MSC:

60G07	General theory of stochastic processes
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G17	Sample path properties
60F17	Functional limit theorems; invariance principles
62E20	Asymptotic distribution theory in statistics
60F15	Strong limit theorems

Keywords:

empirical process theory; Uniform convergence; empirical measures; Convergence in distribution; cadlag functions; Central limit theorems; coupling

Citations:

Zbl 0271.60009