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Markov Chains pp 339-359 | Cite as

Geometric Rates of Convergence

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

Abstract

We have seen in Chapter  11 that a positive recurrent irreducible kernel P on \(\mathsf {X}\times \mathscr {X}\) admits a unique invariant probability measure, say \(\pi \). If the kernel is, moreover, aperiodic, then the iterates of the kernel \(P^n(x,\cdot )\) converge to \(\pi \) in total variation distance for \(\pi \)-almost all \(x\in \mathsf {X}\). Using the characterizations of Chapter  14, we will in this chapter establish conditions under which the rate of convergence is geometric in f-norm, i.e., \(\lim _{n \rightarrow \infty } \delta ^n \left\| P^n(x,\cdot )-\pi \right\| _{f}=0\) for some \(\delta > 1\) and positive measurable function f.

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Randal Douc
    • 1
    Email author
  • 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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