Generalized eigenfunctions of Markov kernels and application to the convergence rate of discrete random walks

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

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $(X_n)_{n\in\N}$ be a Markov chain on a measurable space $\X$ with transition kernel $P$ and let $V:\X\r[1,+\infty)$. Under a weak drift condition, the size of generalized eigenfunctions of $P$ is estimated, where $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then combining this result and quasi-compactness arguments enables us to derive upper bounds for the geometric rate of convergence of $(X_n)_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. Applications to discrete Markov random walks are presented.

Mots clés: Geometric ergodicity; quasi-compactness; Drift condition; Birth-and -Death Markov chains.; Birth-and -Death Markov chains

Domaines: Probabilités [math.PR]

Denis Guibourg, Loïc Hervé, James Ledoux. Generalized eigenfunctions of Markov kernels and application to the convergence rate of discrete random walks. 2012. ⟨hal-00705523v1⟩