Topics in Stochastic Processes  Robert B. Ash 高清英文版

This book contains selected topics in stochastic processes that we believe
can be studied profitably by a reader familiar with basic measure-theoretic
probability. The background is given in " Real Analysis and Probability " by
Robert B. Ash, Academic Press, 1972. A student who has learned this material
from other sources will be in good shape if he feels reasonably comfortable
with infinite sequences of random variables. In particular, a reader who
has studied versions of the strong law of large numbers and the central limit
theorem, as well as basic properties of martingale sequences, should find our
presentation accessible.
We should comment on our choice of topics. In using the tools of
measure-theoretic probability, one is unavoidably operating at a high level
of abstraction. Within this limitation, we have tried to emphasize processes
that have a definite physical interpretation and for which explicit numerical
results can be obtained, if desired. Thus we begin (Chapters 1 and 2) with L2
stochastic processes and prediction theory. Once the underlying mathematical
foundation has been built, results which have been used for many years
by engineers and physicists are obtained. The main result of Chapter 3, the
ergodic theorem, may be regarded as a version of the strong law of large
numbers for stationary stochastic processes. We describe several interesting
applications to real analysis, Markov chains, and information theory.
In Chapter 4 we discuss the sample function behavior of continuous
parameter processes. General properties of martingales and Markovprocesses are given, and one-dimensional Brownian motion is analyzed in
detail. The purpose is to illustrate those concepts and constructions that are
basic in any discussion of continuous parameter processes, and to open the
gate to allow the reader to proceed to more advanced material on Markov
processes and potential theory. In Chapter 5 we use the theory of continuous
parameter processes to develop the Ito stochastic integral and to discuss the
solution of stochastic differential equations. The results are of current interest
in communication and control theory.
The text has essentially three independent units: Chapters 1 and 2;
Chapter 3; and Chapters 4 and 5. The system of notation is standard; for
example, 2.3.1 means Chapter 2, Section 3, Part 1. A reference to "Real
Analysis and Probability " is denoted by RAP.
Problems are given at the end of each section. Fairly detailed solutions
are given to many problems.
We are indebted to Mary Ellen Bock and Ed Perkins for reading the
manuscript and offering many helpful suggestions.
Once again we thank Mrs. Dee Keel for her superb typing, and the staff
of Academic Press for their constant support and encouragement.