## Abstract

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=1}^{n}P^{k}f converges uniformly, then for every x∈S, 1/n ∑_{k=1}^{n}f(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=1}^{n}μ^{k}*f and of 1/n∑_{k=1}^{n}f(X_{k}) via that of Ψ_{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 group G, and let {A_{n}} be a Følner sequence. If for f∈B(G, ∑) m(A_{n})^{-1}∫_{An}f(xt)dm(t) converges uniformly, then 1/n∑_{k=1}^{n}f(X_{k}) converges uniformly, and P_{x} converges P_{x} a.s. for every x∈G.

Original language | English |
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Pages (from-to) | 483-497 |

Number of pages | 15 |

Journal | Journal of Theoretical Probability |

Volume | 7 |

Issue number | 3 |

DOIs | |

State | Published - 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