Stationary and Related Stochastic Processes

Sample Function Properties and Their Applications

Harald Cramér, M. Ross Leadbetter

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486438276
• eBook ISBN: 9780486153353
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This graduate-level text offers a comprehensive account of the general theory of stationary processes, with special emphasis on the properties of sample functions. Assuming a familiarity with the basic features of modern probability theory, the text develops the foundations of the general theory of stochastic processes, examines processes with a continuous-time parameter, and applies the general theory to procedures key to the study of stationary processes. Additional topics include analytic properties of the sample functions and the problem of time distribution of the intersections between a sample function. 1967 edition.

CHAPTER 1

Empirical Background

An important motivation for studying any branch of probability theory arises from its application to practical situations involving random phenomena. That is, the theory provides a mathematical description, or model for, say, a given experiment in which random influences are involved. Such a model provides a basis for making statistical decisions concerning possible courses of action resulting from the experiment. The situation is thus closely akin to that in many other fields of application of mathematics such as, for example, Euclidean geometry, where we have an abstract theory on the one hand and important practical applications on the other.

In this book we shall be concerned primarily with the probabilistic structure of certain types of stochastic processes, or random functions of a variable which, in most practical applications, will signify time. Hence, it is appropriate that we begin with a general discussion of the empirical probability background concerning practical situations which lead to the consideration of such processes.

1.1 RANDOM EXPERIMENTS AND RANDOM VARIABLES

In Chapter 2 we shall be concerned with a proper mathematical foundation for probability theory. It is appropriate here, however, to consider the intuitive concepts of the theory.

The practical situation with which we are concerned involves a random experiment. That is, we consider an experiment E, the outcome of which is determined, at least partially, by some random mechanism whose effect we cannot predict exactly in advance. The word “experiment” is here used in a broad sense; we may be concerned with n tosses of a coin, a measurement of the height of a random individual in a population, or of the lifetime of a light bulb, and so on.

In conducting a random experiment, one is interested in the probabilities that certain events will occur. For example, if A denotes the event that a single toss of an unbiased coin will yield a head, we define P(A), the probability that event A occurs, as having the value

. In any case, the definition of the probability p = P(A) of an event A which may or may not occur when the experiment E is performed, is motivated by the requirement that the proportion of times A occurs in a large number of repetitions of E should eventually, in some reasonable sense, approximate the value p. Probabilities of certain events may often be found by means of combinatorial rules from known probabilities of simpler events; for example, the probability of two heads in two tosses with a coin is

. In more complex situations, a strict definition of the “probability of an event” is best given in measure-theoretic terms involving a mathematical model for the random experiment concerned.

In virtually every kind of random experiment, even the simplest, one is interested in some numerical quantity associated with the experiment. For example, one may be interested in the number of heads in n tosses of a coin, or the total score in n throws of a die. Such a numerical quantity is termed a random variable. The name “random variable” derives from the fact that its value cannot generally be predicted exactly in advance, and repetitions of the experiment do not usually give the same numerical value; if in three tosses of a coin three heads are obtained, there is no guarantee that three further tosses will produce another three heads.

It is random variables which are of primary interest in probability and statistics. Although a random variable is, in general, best defined within a measure-theoretic framework (which will be described in Chapter 2), we note here that the probabilistic behaviour of a random variable ξ is specified by means of its distribution function F(x) where

F(x) = P{ξ ≤ x}.

That is, F(x) is the probability of the event in braces (the event that the experiment yields a value of ξ which does not exceed the real number x). The knowledge of F(x) enables one to obtain the probability that the value of the random variable ξ should lie in a set S of real numbers, for a very wide class of such sets S. In this way, the distribution function uniquely determines the probabilistic properties of interest relative to the random variable ξ.

1.2 FINITE FAMILIES OF RANDOM VARIABLES

In many cases, we are concerned with not just one, but a number of random variables considered simultaneously. For example, if we choose an individual at random from a community and measure his height, weight, and I.Q., we obtain three random variables, ξ₁, ξ₂, and ξ₃.

More generally, we may consider a finite family (ξ₁ ... ξ_n) or vector random variable ξ = (ξ₁ ··· ξ_n), where ξ’s are each random variables. To discuss the probabilistic behaviour of the family, we would like to specify the probability that the point (ξ₁ ··· ξ_n) in n-dimensional space should lie in a given n-dimensional set S. As in the case of a single random variable, this is uniquely specified for a large class of such sets S by the joint distribution function of ξ₁ ··· ξ_n, namely,

F(x₁ ··· x_n) = P(ξ₁ ≤ x_1, ... , ξ_n ≤ x_n).

This distribution function is thus the probability that no ξ_i should exceed the corresponding x_i. Again, a fuller and more precise discussion of the properties of such finite-dimensional distribution functions will be given in the following chapters. Here we simply note that the function F(x₁ ··· x_n) summarizes all the information concerning the probabilities of interest, relative to the joint behaviour of the finite family (ξ₁ ··· ξ_n).

1.3 INFINITE FAMILIES—STOCHASTIC PROCESSES

In this book our main interest will center on the generalization of the preceding cases to include arbitrary families of random variables. That is, we shall consider a family {ξ_t} of random variables where t runs through some index set T. If T consists of a single point, we return to the case of a single random variable, while a finite T-set corresponds to the finite family of the preceding section.

Fig. 1.3.1. Annual mean air temperatures in London from 1763 to 1900.

Fig. 1.3.2. Phase error in phase tracking experiment.

From our point of view, the most interesting cases will be when T consists of the set of integers (or ...

Table of contents

• Title Page
• Copyright Page
• Preface
• Table of Contents
• CHAPTER 1 - Empirical Background
• CHAPTER 2 - Some Fundamental Concepts and Results of Mathematical Probability Theory
• CHAPTER 3 - Foundations of the Theory of Stochastic Processes
• CHAPTER 4 - Analytic Properties of Sample Functions
• CHAPTER 5 - Processes with Finite Second-Order Moments
• CHAPTER 6 - Processes with Orthogonal Increments
• CHAPTER 7 - Stationary Processes
• CHAPTER 8 - Generalizations
• CHAPTER 9 - Analytic Properties of the Sample Functions of Normal Processes
• CHAPTER 10 - “Crossing” Problems and Related Topics
• CHAPTER 11 - Properties of Streams of Crossings
• CHAPTER 12 - Limit Theorems for Crossings
• CHAPTER 13 - Nonstationary Normal Processes Curve Crossing Problems
• CHAPTER 14 - Frequency Detection and Related Topics
• CHAPTER 15 - Some Aspects of the Reliability of Linear Systems
• References
• Index