Journal of Theoretical Probability

, Volume 7, Issue 3, pp 483–497

# Uniform ergodic convergence and averaging along Markov chain trajectories

• Yves Derriennic
• Michael Lin
Article

## Abstract

LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX n be the associated Markov chain.Theorem. LetfB(S, Σ). Then for anyxS we haveP x a.s.
$$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {f(X_k ) \geqslant } \mathop {\underline {\lim } }\limits_{n \to \infty } \mathop {\inf }\limits_{x \in S} \frac{1}{n}\sum\limits_{k = 1}^n {P^k f(x)}$$
and (implied by it) a corresponding inequality for the lim. If 1/n k=1 n P k f converges uniformly, then for everyx∈S, 1/n k=1 n f(X k ) convergesP x a.s.

Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n k=1 n μ k *f and of 1/n k=1 n f(X k ) via that ofΨ n *f(x)=m(A n )−1 An f(xt), where {A n } 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 groupG, and let {A n } be a Følner sequence. If forfB(G, ∑) m(A n )−1 An f(xt)dm(t) converges uniformly, then 1/n k=1 n f(X k ) converges uniformly, andP x convergesP x a.s. for everyxG.

## Key Words

Ergodic theorems strong law of large numbers Markov chains random walks

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