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Markov Chains

  • Christian P. Robert
  • George Casella
Part of the Springer Texts in Statistics book series (STS)

Abstract

In this chapter we introduce fundamental notions of Markov chains and state the results that are needed to establish the convergence of various MCMC algorithms and, more generally, to understand the literature on this topic. Thus, this chapter, along with basic notions of probability theory, will provide enough foundation for the understanding of the following chapters. It is, unfortunately, a necessarily brief and, therefore, incomplete introduction to Markov chains, and we refer the reader to Meyn and Tweedie (1993), on which this chapter is based, for a thorough introduction to Markov chains. Other perspectives can be found in Doob (1953), Chung (1960), Feller (1970, 1971), and Billingsley (1995) for general treatments, and Norris (1997) Nummelin (1984), Revuz (1984), and Resnick (1994) for books entirely dedicated to Markov chains. Given the purely utilitarian goal of this chapter, its style and presentation differ from those of other chapters, especially with regard to the plethora of definitions and theorems and to the rarity of examples and proofs. In order to make the book accessible to those who are more interested in the implementation aspects of MCMC algorithms than in their theoretical foundations, we include a preliminary section that contains the essential facts about Markov chains.

Keywords

Markov Chain Random Walk Invariant Measure Central Limit Theorem Markov Chain Monte Carlo Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Christian P. Robert
    • 1
    • 2
  • George Casella
    • 3
  1. 1.Laboratoire de StatistiqueCREST-INSEEParis Cedex 14France
  2. 2.Dept. de Mathematique UFR des SciencesUniversite de RouenMont Saint Aignan cedexFrance
  3. 3.Biometrics UnitCornell UniversityIthacaUSA

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