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Markov Chains with Stationary Transition Probabilities

  • Textbook
  • © 1960

Overview

Authors:
  1. Kai Lai Chung

    1. Syracuse University, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 104)

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Table of contents (37 chapters)

  1. Front Matter

    Pages II-X
    Download chapter PDF

  2. Discrete parameter

    1. Fundamental defintions

      • Kai Lai Chung
      Pages 1-5

    2. Transition probabilities

      • Kai Lai Chung
      Pages 5-11

    3. Classification of states

      • Kai Lai Chung
      Pages 11-15

    4. Recurrence

      • Kai Lai Chung
      Pages 15-20

    5. Criteria and examples

      • Kai Lai Chung
      Pages 20-26

    6. The main limit theorem

      • Kai Lai Chung
      Pages 26-33

    7. Various complements

      • Kai Lai Chung
      Pages 33-39

    8. Repetitive pattern and renewal process

      • Kai Lai Chung
      Pages 39-43

    9. Taboo probabilities

      • Kai Lai Chung
      Pages 43-52

    10. The generating function

      • Kai Lai Chung
      Pages 52-57

    11. The moments of first entrance time distributions

      • Kai Lai Chung
      Pages 57-65

    12. A random walk example

      • Kai Lai Chung
      Pages 65-71

    13. System theorems

      • Kai Lai Chung
      Pages 71-75

    14. Functionals and associated random variables

      • Kai Lai Chung
      Pages 75-85

    15. Ergodic theorems

      • Kai Lai Chung
      Pages 85-93

    16. Further limit theorems

      • Kai Lai Chung
      Pages 93-106

    17. Almost closed and sojourn sets

      • Kai Lai Chung
      Pages 106-113

  3. Continuous Parameter

    1. Transition matrix: basic properties

      • Kai Lai Chung
      Pages 114-123

    2. Standard transition matrix

      • Kai Lai Chung
      Pages 123-130
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Keywords

About this book

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.

Authors and Affiliations

  • Syracuse University, USA

    Kai Lai Chung

Bibliographic Information

  • Book Title: Markov Chains with Stationary Transition Probabilities

  • Authors: Kai Lai Chung

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-49686-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1960

  • Softcover ISBN: 978-3-642-49408-6Published: 01 January 1960

  • eBook ISBN: 978-3-642-49686-8Published: 08 March 2013

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: X, 278

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes

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