Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks

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 $V:\X\r[1,+\infty)$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then the combination of quasi-compactness arguments with precise analysis of eigen-elements of $P$ allows us to estimate the geometric rate of convergence $\rho_V(P)$ of $\{X_n\}_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. A general procedure to compute $\rho_V(P)$ for discrete Markov random walks with identically distributed bounded increments is specified.

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

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks. 2013. ⟨hal-00705523v2⟩