Markov Chains pp 191-220 | Cite as

Small Sets, Irreducibility, and Aperiodicity

  • Randal Douc
  • Eric Moulines
  • Pierre Priouret
  • Philippe Soulier
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)


So far, we have considered only atomic and discrete Markov chains. When the state space is not discrete, many Markov chains do not admit accessible atoms. Recall that a set C is an atom if each time the chain visits C, it regenerates, i.e., it leaves C under a probability distribution that is constant over C. If the state space does not possess an atom, we may require instead that the chain restart anew from C with some fixed probability (strictly less than one) that is constant over C. Then this property is satisfied by many more Markov chains. Such sets will be called small sets. The purpose of this chapter is to provide the first basic properties of Markov kernels that admit accessible small sets.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Randal Douc
    • 1
  • Eric Moulines
    • 2
  • Pierre Priouret
    • 3
  • Philippe Soulier
    • 4
  1. 1.Département CITITelecom SudParisÉvryFrance
  2. 2.Centre de Mathématiques AppliquéesEcole PloytechniquePalaiseauFrance
  3. 3.Université Pierre et Marie CurieParisFrance
  4. 4.Université Paris NanterreNanterreFrance

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