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

Denis Guibourg (1) , Loïc Hervé (1, 2) , James Ledoux (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes

Let $P$ be a Markov kernel on a measurable space $\mathbb{X}$ and let $V:\mathbb{X}\rightarrow [1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\mathcal{B}_V,\|\cdot\|_V)$ of all the measurable functions $f : \mathbb{X}\rightarrow\mathbb{C}$ such that $\|f\|_V := \sup_{x\in \mathbb{X}} |f(x)|/V(x) < \infty$. 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 $\mathcal{B}_V$, 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.

Mots clés: Markov chain; drift condition; essential spectral radius; convergence rate

Domaines: Probabilités [math.PR]

Denis Guibourg, Loïc Hervé, James Ledoux. Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity. 2012. ⟨hal-00632580v5⟩