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Markov Chains with Stationary Transition Probabilities

  • Kai Lai Chung

Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

Table of contents

  1. Front Matter
    Pages II-X
  2. Discrete parameter

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      Pages 1-5
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      Pages 5-11
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      Pages 11-15
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      Pages 15-20
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      Pages 20-26
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      Pages 26-33
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      Pages 33-39
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      Pages 39-43
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      Pages 43-52
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      Pages 52-57
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      Pages 65-71
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      Pages 71-75
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      Pages 85-93
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      Pages 93-106
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      Pages 106-113
  3. Continuous Parameter

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      Pages 114-123
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      Pages 123-130
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      Pages 130-135
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      Pages 135-143
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      Pages 152-156
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      Pages 168-177
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      Pages 177-182
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      Pages 182-191
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      Pages 196-203
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      Pages 203-209
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  4. Back Matter
    Pages 261-278

About this book

Introduction

The theory of Markov chains, although a special case of Markov processes, is here developed for its own sake and presented on its own merits. In general, the hypothesis of a denumerable state space, which is the defining hypothesis of what we call a "chain" here, generates more clear-cut questions and demands more precise and definitive an­ swers. For example, the principal limit theorem (§§ 1. 6, II. 10), still the object of research for general Markov processes, is here in its neat final form; and the strong Markov property (§ 11. 9) is here always applicable. While probability theory has advanced far enough that a degree of sophistication is needed even in the limited context of this book, it is still possible here to keep the proportion of definitions to theorems relatively low. . From the standpoint of the general theory of stochastic processes, a continuous parameter Markov chain appears to be the first essentially discontinuous process that has been studied in some detail. It is common that the sample functions of such a chain have discontinuities worse than jumps, and these baser discontinuities play a central role in the theory, of which the mystery remains to be completely unraveled. In this connection the basic concepts of separability and measurability, which are usually applied only at an early stage of the discussion to establish a certain smoothness of the sample functions, are here applied constantly as indispensable tools.

Keywords

Markov chain Markov process Markov property Moment Parameter Random Walk Random variable probability probability theory stochastic processes

Authors and affiliations

  • Kai Lai Chung
    • 1
  1. 1.Syracuse UniversityUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-49686-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1960
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-49408-6
  • Online ISBN 978-3-642-49686-8
  • Buy this book on publisher's site