General irreducible Markov chains and non-negative operators


General irreducible Markov chains and non-negative operators

Esa Nummelin

(Cambridge tracts in mathematics, 83)

Cambridge University Press, 1984

  • : hard

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Bibliography: p. 148-153

Includes index



The purpose of this book is to present the theory of general irreducible Markov chains and to point out the connection between this and the Perron-Frobenius theory of nonnegative operators. The author begins by providing some basic material designed to make the book self-contained, yet his principal aim throughout is to emphasize recent developments. The technique of embedded renewal processes, common in the study of discrete Markov chains, plays a particularly important role. The examples discussed indicate applications to such topics as queueing theory, storage theory, autoregressive processes and renewal theory. The book will therefore be useful to researchers in the theory and applications of Markov chains. It could also be used as a graduate-level textbook for courses on Markov chains or aspects of operator theory.


  • Preface
  • 1. Preliminaries
  • 2. Irreducible kernels
  • 3. Transience and recurrence
  • 4. Embedded renewal processes
  • 5. Positive and null recurrence
  • 6. Total variation limit theorems
  • 7. Miscellaneous limit theorems for Harris recurrent Markov chains
  • Notes and comments
  • List of symbols and notation
  • Bibliography
  • Index.

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