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Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces. (English) Zbl 1186.37014

The aim of the present paper is to study quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces. In particular the author managed to prove that if \(P\) is a Markov kernel on a measurable space \(E\) with countably generated \(\sigma\)-algebra, \(w: E\to[1,+\propto[\) such that \(Pw\leq Cw\) with \(C\geq 0\) and \(B_w\) the space of measurable functions on \(E\) satisfying \(\| f\|_w= \sup\{w(x)^{-1}|f(x)|, x\in E\}<+\propto\) then \(P\) is quasi-compact on the space \((B_w,\|.\|_w)\), if and only if, for all \(f\in B_w\), \(({1\over n}\sum^n_{k=1} P^k f)_n\) contains a subsequence converging in \(B_w\) to \(\Pi f=\sum^d_{i=1} \mu_i(f)v_i\), where the \(v_i\)’s are nonnegative bounded measurable functions on \(E\) and the \(\mu_i\)’s are probability distributions on \(E\).
Moreover in the case that the space of \(P\)-invariant functions in \(B_w\) is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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References:

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