Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition

A Note on Invariant Finitely Additive Measures

We show that under certain general conditions any finitely additive measure which is defined for all subsets of a set X and is invariant under the action of a group G acting on X is concentrated on a G-invariant subset Y on which the G-action factors to that of an amenable group. The result is then applied to prove a conjecture of S. Wagon about finitely additive measures on spheres. It is well...

Drift conditions and invariant measures for Markov chains

We consider the classical Foster–Lyapunov condition for the existence of an invariant measure for a Markov chain when there are no continuity or irreducibility assumptions. Provided a weak uniform countable additivity condition is satis/ed, we show that there are a /nite number of orthogonal invariant measures under the usual drift criterion, and give conditions under which the invariant measur...

Representing Finitely Additive Invariant Probabilities

Geometric Convergence of Markov Chains through a Local Doeblin Condition B. Delyon

An operator-theoretical approach is used to prove that the transition kernel of a Markov chain has, under a local Doeblin condition, a nite number of eigenvalues of modulus larger than some < 1. This leads to a representation of the transition kernel P as a nite sum of rank 1 operators P i , each of them associated with a periodic or absorbing set, plus a contracting operator T such that P i T ...

On finitely additive vector measures.

In this paper we extend the well-known Vitali-Hahn-Saks and Nikodým theorems for measures to finitely additive vector-valued set functions.