Truncated Markov chains and V-geometric ergodicity via weak perturbation theory

Loïc Hervé (1) , James Ledoux (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let P be a Markov kernel on a metric space X and let V be a function from X to [1,+\infty[. This paper provides explicit connections between the V-geometrical ergodicity of P and that of its truncated kernels. Furthermore an explicit bound on the total variation distance between their invariant probability measures is provided. The proofs are based on the Keller-Liverani perturbation theorem. which requires an accurate control of the essential spectral radius of both P and the truncated kernels as linear operators on the V-weighted supremum Banach space B_1. Consequently, a part of this paper is devoted to the derivation of computable bounds on the essential spectral radius on B_1 of a general Markov kernel from standard drift conditions.

Mots clés: quasi-compact operator; drift condition

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Truncated Markov chains and V-geometric ergodicity via weak perturbation theory. 2012. ⟨hal-00704689v1⟩