Stochastic Processes

This introductory textbook explains how and why probability models are applied to scientific fields such as medicine, biology, physics, oceanography, economics, and psychology to solve problems about stochastic processes. It does not just show how a problem is solved but explains why by formulating questions and first steps in the solutions. Stochastic Processes is ideal for a course aiming to give examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models. It introduces the methods of probability model building and provides the reader with mathematically sound techniques as well as the ability to further study the theory of stochastic processes. Originally published in 1962, this was the first comprehensive survey of stochastic processes requiring only a minimal background in introductory probability theory and mathematical analysis. Stochastic Processes continues to be unique, with many topics and examples still not discussed in other textbooks.

Emanuel Parzen is Distinguished Professor of Statistics at Texas AandM University

Preface to the Classics Edition	p. xiii
Preface	p. xvii
Role of the Theory of Stochastic Processes
Statistical physics	p. 1
Stochastic models for population growth	p. 2
Communication and control	p. 2
Management science	p. 4
Time series analysis	p. 5
Random Variables and Stochastic Processes
Random variables and probability laws	p. 8
Describing the probability law of a stochastic process	p. 22
The Wiener process and the Poisson process	p. 26
Two-valued processes	p. 35
Conditional Probability and Conditional Expectation
Conditioning by a discrete random variable	p. 42
Conditioning by a continuous random variable	p. 51
Properties of conditional expectation	p. 62
Normal Processes and Covariance Stationary Processes
The mean value function and covariance kernel of a stochastic process	p. 66
Stationary and evolutionary processes	p. 69
Integration and differentiation of stochastic processes	p. 78
Normal processes	p. 88
Normal processes as limits of stochastic processes	p. 97
Harmonic analysis of stochastic processes	p. 103
Counting Processes and Poisson Processes
Axiomatic derivations of the Poisson process	p. 118
Non-homogeneous, generalized, and compound Poisson processes	p. 124
Inter-arrival times and waiting times	p. 132
The uniform distribution of waiting times of a Poisson process	p. 139
Filtered Poisson processes	p. 144
Renewal Counting Processes
Examples of renewal counting processes	p. 160
The renewal equation	p. 170
Limit theorems for renewal counting processes	p. 180
Markov Chains: Discrete Parameter
Formal definition of a Markov process	p. 188
Transition probabilities and the Chapman-Kolmogorov equation	p. 193
Decomposition of Markov chains into communicating classes	p. 208
Occupation times and first passage times	p. 211
Recurrent and non-recurrent states and classes	p. 221
First passage and absorption probabilities	p. 226
Mean absorption, first passage, and recurrence times	p. 238
Long-run and stationary distributions	p. 247
Limit theorems for occupation times	p. 265
Limit theorems for transition probabilities of a finite Markov chain	p. 270
The interchange of limiting processes	p. 273
Markov Chains: Continuous Parameter
Limit theorems for transition probabilities of a continuous parameter Markov chain	p. 276
Birth and death processes and their application to queueing theory	p. 278
Kolmogorov differential equations for the transition probability functions	p. 288
Two-state Markov chains and pure birth processes	p. 293
Non-homogeneous birth and death processes	p. 299
References	p. 307
Author Index	p. 314
Subject Index	p. 316
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