Convergence of stochastic processes. (English) Zbl 0544.60045

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


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


Zbl 0271.60009