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Unformatted text preview: Probability, Random Processes, and Ergodic Properties November 3, 2001 ii Probability, Random Processes, and Ergodic Properties Robert M. Gray Information Systems Laboratory Department of Electrical Engineering Stanford University iv c 1987 by Springer Verlag, 2001 revision by Robert M. Gray. v This book is aﬀectionately dedicated to Elizabeth Dubois Jordan Gray and to the memory of R. Adm. Augustine Heard Gray, U.S.N. 18881981 Sara Jean Dubois and William “Billy” Gray 17501825 vi Preface History and Goals This book has been written for several reasons, not all of which are academic. This material was for many years the ﬁrst half of a book in progress on information and ergodic theory. The intent was and is to provide a reasonably selfcontained advanced treatment of measure theory, probability theory, and the theory of discrete time random processes with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary. The intended audience was mathematically inclined engineering graduate students and visiting scholars who had not had formal courses in measure theoretic probability. Much of the material is familiar stuﬀ for mathematicians, but many of the topics and results have not previously appeared in books. The original project grew too large and the ﬁrst part contained much that would likely bore mathematicians and discourage them from the second part. Hence I ﬁnally followed a suggestion to separate the material and split the project in two. The original justiﬁcation for the present manuscript was the pragmatic one that it would be a shame to waste all the eﬀort thus far expended. A more idealistic motivation was that the presentation had merit as ﬁlling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash’s excellent text [1], which was likely inﬂuenced by his original training as an electrical engineer. Still, even that text devotes little eﬀort to ergodic theorems, perhaps the most fundamentally important family of results for applying probability theory to real problems. In addition, there are many other special topics that are given little space (or none at all) in most texts on advanced probability and random processes. Examples of topics developed in more depth here than in most existing texts are the following: Random processes with standard alphabets We develop the theory of standard spaces as a model of quite general process alphabets. Although not as general (or abstract) as often considered by probability theorists, standard spaces have useful structural properties that...
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 Fall '10
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 Probability theory, Random Processes

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