Functional analysis for probability and stochastic processes : an introduction /

Bobrowski (Lublin Univ. of Technology) lucidly presents this material for those with good backgrounds in functional analysis but little knowledge of probability, and for statisticians with no knowledge of functional analysis. Readers need a good knowledge of measure theory, some exposure to solving ordinary differential equations, and some knowledge of abstract algebra and topology--all briefly sketched in chapter 1. Other chapters discuss linear spaces, Banach spaces, and the space of bounded linear operators; conditional expectations and their properties; Brownian motion, Wiener's proof of the existence of Brownian motion by using Hilbert Space theory, and the Ito integral; various modes of convergence of probability measures, the notion of a Banach limit, dual spaces, and compact sets; Gelfand transforms; and Markov processes: Levy processes (a particular type of Markov process) and the Hille-Yosida theorem, according to which there is a 1-1 correspondence between Markov processes and a class of linear operators--the class of generators of corresponding semi groups. Three appendixes contain bibliographical notes, solutions and hints for exercises, and a list of some commonly used notations. An attractive feature is the numerous solved examples. A good addition to the statistical literature. Useful index. ^BSumming Up: Recommended. Graduate students; faculty. D. V. Chopra Wichita State University