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Stationary and Related Stochastic Processes Sample Function Properties and Their Applications Harald Cramér M. R. Leadbetter 4

Copyright Copyright © 1967 by John Wiley & Sons, Inc. Copyright © renewed 1995 by M. R. Leadbetter and the Estate of Harald Cramér All rights reserved. Bibliographical Note This Dover edition, first published in 2004, is an unabridged republication of the work originally published in 1967 by John Wiley & Sons, Inc. New York. 9780486153353 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501 5

Preface This book is an outcome of a collaboration between the authors, under the auspices of the Research Triangle Institute (RTI) of Durham, North Carolina, through contracts with the National Aeronautics and Space Administration. Starting in 1962, our joint work was originally concerned with certain reliability problems, which were found to be intimately connected with the properties of the trajectories (or sample functions) of stationary stochastic processes. After we had reached some preliminary results, it was suggested by the Senior Scientific Advisory Committee of the RTI that we should follow up our work and write a joint monograph on these and some related problems. When trying to work out a possible programme for such a book, we soon came to the conclusion that the most desirable plan would be to include an account of the general theory of stationary processes, with special emphasis on the properties of their sample functions. It is well known that some of these properties are important also in other fields of application, such as communication engineering. Accordingly, we decided to make the discussion of the sample functions sufficiently broad to cover as many as possible of the properties relevant for such applications. The book is written for a reader assumed to have a working knowledge of the basic features of modern probability theory. However, some fundamental concepts and propositions of this theory are briefly reviewed in an introductory chapter (Chapter 2). The foundations of the general theory of stochastic processes are then 6

developed, with special emphasis on processes with a continuous-time parameter (Chapter 3). The analytic properties of the trajectories (or sample functions), such as continuity, differentiability etc., are studied in some detail (Chapter 4). The general theory is then applied to certain classes of processes important as tools for the study of stationary processes (Chapters 5 and 6). The main part of the book is concerned with the theory and applications of stationary processes. Their spectral representation is deduced by methods of Hilbert space geometry introduced in a previous chapter, analytic properties of the sample functions are studied, and proofs of some basic ergodic theorems are given (Chapter 7). On the other hand, problems of prediction and filtering, of which excellent accounts are available elsewhere, are only briefly discussed. Certain generalizations are treated in a separate chapter (Chapter 8). In the important case of normal (Gaussian) stationary processes, the conditions for continuity, etc., of the sample functions take a particularly simple form; this form is thoroughly studied in Chapter 9. For this class of processes, the problem of the time distribution of the intersections between a sample function and a given constant level, or a given curve, has recently attracted a considerable interest. Problems of this type are extensively discussed, and this is believed to be the first account in book form of much recent work by American, Soviet Russian, and other authors. Several results believed to be new are given in this connection (Chapters 10 to 13). Various applications to problems of frequency detection and reliability are given (Chapters 14 to 15). We have endeavoured to give complete and rigorous mathematical proofs of all results judged to be important from the point of view of the applications. At the same time an attempt has been made to reduce the mathematical difficulties involved as far as possible. Our most sincere thanks are due to the National Aeronautics and Space Administration and to the Research Triangle Institute for their support of our joint work, to Professors S. S. Wilks and R. F. Drenick for encouraging us to write this book, and to Dr. J. D. Cryer for his painstaking and critical reading of the entire manuscript. As will be seen, much of the content of the later chapters leans heavily on the pioneering work of S. O. Rice. It is a pleasure to express our gratitude to Dr. Rice and Dr. David Slepian for very helpful and 7

stimulating conversations. Finally, we owe a very special debt of gratitude to Dr. Gertrude M. Cox. It was largely through the efforts of Dr. Cox that our collaboration became possible, and we very much appreciated her continued interest and enthusiasm throughout the entire project. H. C., M. R. L. July 1966 8

Table of Contents Title Page Copyright Page Preface 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 9

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 10

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