Uniform ergodic convergence and averaging along Markov chain trajectories

Let P(x, A) be a transition probability on a measurable space (S, Σ) and let X_n be the associated Markov chain. Theorem. Let f∈B(S, Σ). Then for any x∈S we have P_x a.s. {Mathematical expression} and (implied by it) a corresponding inequality for the lim. If 1/n∑_k=1ⁿP^kf converges uniformly, then for every x∈S, 1/n ∑_k=1ⁿf(X_k) converges P_x a.s. Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n∑_k=1ⁿμ^k*f and of 1/n∑_k=1ⁿf(X_k) via that of Ψ_n*f(x)=m(A_n)^-1∫_Anf(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 group G, and let {A_n} be a Følner sequence. If for f∈B(G, ∑) m(A_n)^-1∫_Anf(xt)dm(t) converges uniformly, then 1/n∑_k=1ⁿf(X_k) converges uniformly, and P_x converges P_x a.s. for every x∈G.