Uniform ergodic convergence and averaging along Markov chain trajectories

Yves Derriennic, Michael Lin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let P(x, A) be a transition probability on a measurable space (S, Σ) and let Xn be the associated Markov chain. Theorem. Let f∈B(S, Σ). Then for any x∈S we have Px a.s. {Mathematical expression} and (implied by it) a corresponding inequality for the lim. If 1/n∑k=1nPkf converges uniformly, then for every x∈S, 1/n ∑k=1nf(Xk) converges Px a.s. Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n∑k=1nμk*f and of 1/n∑k=1nf(Xk) via that of Ψn*f(x)=m(An)-1Anf(xt), where {An} is a Følner sequence, in the following cases: (i)f is left uniformly continuous (ii) μ is spread out (iii)G is Abelian. Non-Abelian Example: Let μ be adapted and spread-out on a nilpotent σ-compact locally compact group G, and let {An} be a Følner sequence. If for f∈B(G, ∑) m(An)-1Anf(xt)dm(t) converges uniformly, then 1/n∑k=1nf(Xk) converges uniformly, and Px converges Px a.s. for every x∈G.

Original languageEnglish
Pages (from-to)483-497
Number of pages15
JournalJournal of Theoretical Probability
Volume7
Issue number3
DOIs
StatePublished - 1 Jul 1994

Keywords

  • Ergodic theorems
  • Markov chains
  • random walks
  • strong law of large numbers

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics (all)
  • Statistics, Probability and Uncertainty

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