Uniform ergodic convergence and averaging along Markov chain trajectories

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. $$mathop {underline {lim } }limits_{n to infty } frac{1}{n}sumlimits_{k = 1}^n {f(X_k ) geqslant } mathop {underline {lim } }limits_{n to infty } mathop {inf }limits_{x in S} frac{1}{n}sumlimits_{k = 1}^n {P^k f(x)} $$ 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 forf∈B(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 everyx∈G.