Markov Chains & Stochastic Stability

Markov Chains & Stochastic Stability - Preface Books...

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Unformatted text preview: Preface Books are individual and idiosyncratic. In trying to understand what makes a good book, there is a limited amount that one can learn from other books; but at least one can read their prefaces, in hope of help. Our own research shows that authors use prefaces for many different reasons. Prefaces can be explanations of the role and the contents of the book, as in Chung [49] or Revuz [223] or Nummelin [202]; this can be combined with what is almost an apology for bothering the reader, as in Billingsley [25] or C inlar [40]; prefaces can describe the mathematics, as in Orey [208], or the importance of the applications, as in Tong [267] or Asmussen [10], or the way in which the book works as a text, as in Brockwell and Davis [32] or Revuz [223]; they can be the only available outlet for thanking those who made the task of writing possible, as in almost all of the above (although we particularly like the familial gratitude of Resnick [222] and the dedication of Simmons [240]); they can combine all these roles, and many more. This preface is no different. Let us begin with those we hope will use the book. Who wants this stuff anyway? This book is about Markov chains on general state spaces: sequences n evolving randomly in time which remember their past trajectory only through its most recent value. We develop their theoretical structure and we describe their application. The theory of general state space chains has matured over the past twenty years in ways which make it very much more accessible, very much more complete, and (we at least think) rather beautiful to learn and use. We have tried to convey all of this, and to convey it at a level that is no more dicult than the corresponding countable space theory. The easiest reader for us to envisage is the long-suffering graduate student, who is expected, in many disciplines, to take a course on countable space Markov chains. Such a graduate student should be able to read almost all of the general space theory in this book without any mathematical background deeper than that needed for studying chains on countable spaces, provided only that the fear of seeing an in-tegral rather than a summation sign can be overcome. Very little measure theory or analysis is required: virtually no more in most places than must be used to define transition probabilities. The remarkable Nummelin-Athreya-Ney regeneration tech-nique, together with coupling methods, allows simple renewal approaches to almost all of the hard results. Courses on countable space Markov chains abound, not only in statistics and mathematics departments, but in engineering schools, operations research groups and ii even business schools. This book can serve as the text in most of these environments for a one-semester course on more general space applied Markov chain theory, pro-vided that some of the deeper limit results are omitted and (in the interests of a fourteen week semester) the class is directed only to a subset of the examples, con-...
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This note was uploaded on 11/19/2010 for the course ECE 376B taught by Professor Tomcover during the Spring '05 term at Stanford.

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Markov Chains & Stochastic Stability - Preface Books...

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