Uniform ergodic convergence and averaging along Markov chain trajectories

Abstract

LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX _n be the associated Markov chain.Theorem. Letf∈B(S, Σ). Then for anyx∈S 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∑ ⁿ _k=1 P ^k f converges uniformly, then for everyx∈S, 1/n ∑ ⁿ _k=1 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∑ ⁿ _k=1 μ^k*f and of 1/n∑ ⁿ _k=1 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 forf∈B(G, ∑) m(A _n )⁻¹∫ _An f(xt)dm(t) converges uniformly, then 1/n∑ ⁿ _k=1 f(X _k ) converges uniformly, andP _x convergesP _x a.s. for everyx∈G.