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  • © 2000

Introduction to Markov Chains

With Special Emphasis on Rapid Mixing

Authors:

  • Aktuelles aus der Theorie der Markovketten

Part of the book series: Advanced Lectures in Mathematics (ALM)

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Table of contents (22 chapters)

  1. Front Matter

    Pages i-ix
  2. Finite Markov chains (the background)

    1. Front Matter

      Pages 1-3
    2. Markov chains: how to start?

      • Ehrhard Behrends
      Pages 4-11
    3. Examples of Markov chains

      • Ehrhard Behrends
      Pages 12-18
    4. How linear algebra comes into play

      • Ehrhard Behrends
      Pages 19-22
    5. Transient states

      • Ehrhard Behrends
      Pages 36-41
    6. An analytical lemma

      • Ehrhard Behrends
      Pages 42-47
    7. Irreducible Markov chains

      • Ehrhard Behrends
      Pages 48-60
    8. Notes and remarks

      • Ehrhard Behrends
      Pages 61-65
  3. Rapidly mixing chains

    1. Front Matter

      Pages 67-70
    2. Perron-Frobenius theory

      • Ehrhard Behrends
      Pages 71-76
    3. Rapid mixing: a first approach

      • Ehrhard Behrends
      Pages 77-90
    4. Conductance

      • Ehrhard Behrends
      Pages 91-101
    5. Stopping times and the strong Markov property

      • Ehrhard Behrends
      Pages 102-107
    6. Coupling methods

      • Ehrhard Behrends
      Pages 108-120
    7. Strong uniform times

      • Ehrhard Behrends
      Pages 121-127
    8. Notes and remarks

      • Ehrhard Behrends
      Pages 168-169

About this book

The aims of this book are threefold:
We start with a naive description of a Markov chain as a memoryless random walk on a finite set. This is complemented by a rigorous definition in the framework of probability theory, and then we develop the most important results from the theory of homogeneous Markov chains on finite state spaces.
Chains are called rapidly mixing if all of the associated walks, regardles of where they started, behave similarly already after comparitively few steps: it is impossible from observing the chain to get information on the starting position or the number of steps done so far. We will thoroughly study methods which have been proposed in the last decades to investigate this phenomenon.
A number of examples will be studied to indicate how the methods treated in this book can be applied.

Authors and Affiliations

  • Institut für Mathematik I, Freie Universität Berlin, Berlin, Germany

    Ehrhard Behrends

About the author

Prof. Dr. Ehrhard Behrends ist am Fachbereich Mathematik der Freien Universität Berlin tätig.

Bibliographic Information

  • Book Title: Introduction to Markov Chains

  • Book Subtitle: With Special Emphasis on Rapid Mixing

  • Authors: Ehrhard Behrends

  • Series Title: Advanced Lectures in Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-90157-6

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Fachmedien Wiesbaden 2000

  • Softcover ISBN: 978-3-528-06986-5Published: 29 November 1999

  • eBook ISBN: 978-3-322-90157-6Published: 08 July 2014

  • Series ISSN: 0932-7134

  • Series E-ISSN: 2512-7039

  • Edition Number: 1

  • Number of Pages: IX, 234

  • Number of Illustrations: 6 b/w illustrations

  • Topics: Mathematics, general, Engineering, general

Buy it now

Buying options

eBook USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access