Stochastic Processes

Emanuel Parzen

• Publisher: Dover Publications
• Year: 2015
• Print ISBN: 9780486796888
• eBook ISBN: 9780486804712
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Well-written and accessible, this classic introduction to stochastic processes and related mathematics is appropriate for advanced undergraduate students of mathematics with a knowledge of calculus and continuous probability theory. The treatment offers examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models, and it develops the methods of probability model-building.
Chapter 1 presents precise definitions of the notions of a random variable and a stochastic process and introduces the Wiener and Poisson processes. Subsequent chapters examine conditional probability and conditional expectation, normal processes and covariance stationary processes, and counting processes and Poisson processes. The text concludes with explorations of renewal counting processes, Markov chains, random walks, and birth and death processes, including examples of the wide variety of phenomena to which these stochastic processes may be applied. Numerous examples and exercises complement every section.

CHAPTER

1

Random variables and stochastic processes

PROBABILITY THEORY is regarded in this book as the study of mathematical models of random phenomena. A random phenomenon is defined as an empirical phenomenon that obeys probabilistic, rather than deterministic, laws.

A random phenomenon that arises through a process (for example, the motion of a particle in Brownian motion, the growth of a population such as a bacterial colony, the fluctuating current in an electric circuit owing to thermal noise or shot noise, or the fluctuating output of gasoline in successive runs of an oil-refining mechanism) which is developing in time in a manner controlled by probabilistic laws is called a stochastic process.†

For reasons indicated in the Introduction, from the point of view of the mathematical theory of probability a stochastic process is best defined as a collection {X(t), t

T} of random variables. (The Greek letter e is read “belongs to” or “varying in.”) The set T is called the index set of the process. No restriction is placed on the nature of T. However, two important cases are when T = {0, ± 1, ± 2, · · · } or T = {0, 1, 2, · · · }, in which case the stochastic process is said to be a discrete parameter process, or when T = {t: − ∞ < t < ∞} or T = {t: ≥ 0}, in which case the stochastic process is said to be a continuous parameter process.

This chapter discusses the precise definition of random variables and stochastic processes that will be employed in this book. It also introduces two stochastic processes, the Wiener process and the Poisson process, that play a central role in the theory of stochastic processes.

1-1RANDOM VARIABLES AND PROBABILITY LAWS

Intuitively, a random variable X is a real-valued quantity which has the property that for every set B of real numbers there exists a probability, denoted by P[X is in B], that X is a member of B. Thus X is a variable whose values are taken randomly (that is, in accord with a probability distribution). In the theory of probability, a random variable is defined as a function on a sample description space. By employing such a definition, one is able to develop a calculus of random variables studying the characteristics of random variables generated, by means of various analytic operations, from other random variables.†

In order to give a formal definition of the notion of a random variable we must first introduce the notions of

(i) a sample description space,

(ii) an event,

(iii) a probability function.

The sample description space S of a random phenomenon is the space of descriptions of all possible outcomes of the phenomenon.

An event is a set of sample descriptions. An event E is said to occur if and only if the observed outcome of the random phenomenon has a sample description in E.

It should be noted that, for technical reasons, one does not usually permit all subsets of S to be events. Rather as the family

of events, one adopts a family

of subsets of S which has the following properties:

(i) S belongs to

.

(ii) To

belongs the complement E^c of any set E belonging to

.

(iii) To

belongs the union

of any...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Role of The Theory of Stochastic Processes
• 1 Random Variables and Stochastic Processes
• 2 Conditional Probability and Conditional Expectation
• 3 Normal Processes and Covariance Stationary Processes
• 4 Counting Processes and Poisson Processes
• 5 Renewal Counting Processes
• 6 Markov Chains: Discrete Parameter
• 7 Markov Chains: Continuous Parameter
• References
• Author Index
• Subject Index