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

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 $V:\mathbb{X}\rightarrow [1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\widehat{P}_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\widehat{P}_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\widehat{P}_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\mathbb{X}:=\mathbb{N}$.

Mots clés: rate of convergence; essential spectral radius; drift condition; quasi-compactness; truncation of discrete kernels.; truncation of discrete kernels

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Approximating Markov chains and V-geometric ergodicity via weak perturbation theory. Stochastic Processes and their Applications, 2014, 124 (1), pp.613-638. ⟨10.1016/j.spa.2013.09.003⟩. ⟨hal-00704689v7⟩