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

Abstract

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.

45 pages

References in corpus (3)

• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Limit theorems for stationary Markov processes with L2-spectral gap
• Quasi-compactness and absolutely continuous kernels, applications to Markov chains

Cited by in corpus (1)

• Approximating Markov chains and V-geometric ergodicity via weak perturbation theory