Introduction to Markov Chains

With Special Emphasis on Rapid Mixing

  • Ehrhard Behrends

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

Table of contents

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

    1. Front Matter
      Pages 1-3
    2. Ehrhard Behrends
      Pages 4-11
    3. Ehrhard Behrends
      Pages 12-18
    4. Ehrhard Behrends
      Pages 19-22
    5. Ehrhard Behrends
      Pages 36-41
    6. Ehrhard Behrends
      Pages 42-47
    7. Ehrhard Behrends
      Pages 48-60
    8. Ehrhard Behrends
      Pages 61-65
  3. Rapidly mixing chains

    1. Front Matter
      Pages 67-70
    2. Ehrhard Behrends
      Pages 71-76
    3. Ehrhard Behrends
      Pages 77-90
    4. Ehrhard Behrends
      Pages 91-101
    5. Ehrhard Behrends
      Pages 102-107
    6. Ehrhard Behrends
      Pages 108-120
    7. Ehrhard Behrends
      Pages 121-127
    8. Ehrhard Behrends
      Pages 168-169

About this book

Introduction

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 some methods
which have been proposed in the last decades to investigate this
phenomenon.
-- Several examples will be studied to indicate how
the methods treated in this book can be applied.

Besides the investigation of general chains the book contains
chapters which are concerned with eigenvalue techniques, conductance,
stopping times, the strong Markov property, couplings, strong uniform
times, Markov chains on arbitrary finite groups (including a
crash-course in harmonic analysis), random generation and counting,
Markov random fields, Gibbs fields, the Metropolis sampler, and simulated annealing. Readers are invited to solve as many as possible of the 170 exercises.

The book is self-contained, emphasis is laid on an
extensive motivation of the ideas rather than on an encyclopaedic
account.
It can be mastered by everyone who has a background in elementary
probability theory and linear algebra.

The author is professor of mathematics at Free University of Berlin, his fields of research are functional analysis and probability theory.







Keywords

Counting Finite Markowsche Kette Mathematische Statistik Wahrscheinlichkeitstheorie calculus

Authors and affiliations

  • Ehrhard Behrends
    • 1
  1. 1.Institut für Mathematik IFreie Universität BerlinBerlinGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-322-90157-6
  • Copyright Information Springer Fachmedien Wiesbaden 2000
  • Publisher Name Vieweg+Teubner Verlag, Wiesbaden
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-528-06986-5
  • Online ISBN 978-3-322-90157-6
  • Series Print ISSN 0932-7134
  • About this book