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 $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometrical ergodicity of $P$ and that of its truncated and augmented kernels $P_k$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $P_k$ and conversely. Furthermore, explicit bounds for the total variation distance between the invariant probability measures of $P$ and $P_k$ are presented. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of both linear operators $P$ and $P_k$ on usual weighted supremum spaces. To that effect, computable bounds for the essential spectral radius of a general Markov kernel on this space are derived in terms of standard drift conditions.

Mots clés: drift condition; rate of convergence; approximations of Markov kernels; essential spectral radius; drift condition.

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Truncated Markov chains and V-geometric ergodicity via weak perturbation theory. 2012. ⟨hal-00704689v2⟩