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Stationary and Related Stochastic Processes: Sample Functions Properties and Their Applications

Harald Cramér and M. R. Leadbetter
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Harald Cramér had a very diverse career. He started with a thesis on Dirichlet series prepared under the direction of Marcel Riesz, got involved with analytic number theory, and moved on to probability. For a considerable part of his academic life he was professor of actuarial science. After a distinguished career at Stockholm University that included several administrative positions, he retired. He then devoted the next twenty years to research and spent a good deal of time at universities in the U.S. It was during this time that he met his collaborator and co-author and began work on this book.

This book is a monograph on stochastic processes. According to the preface, the work was motivated by reliability problems associated with properties of trajectories of stationary stochastic processes that were encountered in work done for NASA. The text is written at the level of a graduate school course, but it has no exercises and may be more suitable as a reference than as a textbook.

The authors provide a general introduction to stochastic processes with a particular emphasis on stationary processes and their associated sample functions. Strictly stationary stochastic processes have underlying probability distributions that do not change in time, so their mean, variance, and autocorrelation are constant. (The authors also consider wide-sense or covariance-stationary processes.) A sample function is a realization of a random process. Each realization results in a different sample function, so one important goal is to characterize general analytical properties of the sample functions for a given stochastic process.

The book begins with a discussion of the empirical background of the authors’ work that connects the largely theoretical material that follows with some practical examples of the stationary processes of the kind that they would later analyze. After introductory material on fundamental concepts and results from probability, the authors introduce the basic elements of the general theory of stochastic processes. Several chapters treat the central topic of sample functions. Their analytic properties (continuity, differentiability, etc.) are treated in some detail. Crossing problems — determining whether, how and how often the values of sample functions cross some threshold — also get extensive attention.

Toward the end of the book the authors discuss applications of spectral analysis to the reliability of linear systems. They provide a nice example of a specific application to guidance systems.

The lemmas, theorems and corollaries in the book are carefully stated and proved, but they are often not typographically distinguished from the rest of the text — they are only italicized. Since many other items (examples, new terms, section headings) are also italicized, it can be difficult to find theorems easily. An interested reader might need to do some judicious underlining or highlighting to help keep track important results.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. Empirical Background
2. Some Fundamental Concepts and Results of Mathematical Probability Theory
3. Foundations of the Theory of Stochastic Processes
4. Analytical Properties of Sample Functions
5. Processes with Finite Second-Order Moments
6. Processes with Orthogonal Increments
7. Stationary Processes
8. Generalizations
9. Analytical Properties of the Sample Functions of Normal Processes
10. Crossing" Problems and Related Topics
11. Properties of Streams of Crossings
12. Limit Theorems for Crossings
13. Nonstationary Normal Processes. Curve Crossing Problems
14. Frequency Detections and Related Topics
15. Some Aspects of the Reliability of Linear Systems