Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Details Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

2011-10-14

arxiv.org/abs/1110.3240v3

Mathematics

Probability

45 pages Include a detailed discussion on a paper by L. Wu

Scientific paper

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.

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