Stopping times and directed processes
By: G.A. Edgar and Louis Sucheston.
Contributors: Sucheston, Louis.
Series: Encyclopedia of mathematics and its applications ;; v. 47
Publisher: Cambridge University Press
Published: 1992
Place of publication: Cambridge [England]; New York, NY, USA
Language: eng
Number of pages: 428
Subjects: Convergence.; Probabilities.; Martingales (Mathematics)
Description: "In this book the technique of stopping times is applied to prove convergence theorems for stochastic processes - in particular processes indexed by direct sets - and in sequential analysis. Applications of convergence theorems are seen in probability, analysis, and ergodic theory." "Almost everywhere, convergence and stochastic convergence of processes indexed by a directed set are studied, and solutions are given for problems left open in Krickeberg's theory for martingales and submartingales. The rewording of Vitali covering conditions in terms of stopping times establishes connections with the theory of stochastic processes and derivation. A study of martingales yields laws of large numbers for martingale differences, with application to "star-mixing" processes. Convergence of processes taking values in Banach spaces is related to geometric properties of these spaces. There is a self-contained section on operator ergodic theorems: the superadditive, Chacon-Ornstein, and Chacon theorems." "A recurrent theme of the book is the unification of martingale and ergodic theorems. One example is the use of a "three-function inequality," which is basic in all the one and many parameter results. A general principle is proved showing that in both theories all the multiparameter convergence theorems follow from one-parameter maximal and convergence theorems." "Requiring only a knowledge of basic measure theory, this book will be a valuable reference for students and researchers in probability theory, analysis, and statistics."--BOOK JACKET.
Notes: Includes bibliographical references (p. [407]-417) and indexes.