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

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 $\{X_n\}_{n\in\mathbb{N}}$ be a Markov chain on a measurable space $\mathbb{X}$ with transition kernel $P$ and let $V:\mathbb{X}\rightarrow [1,+\infty)$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space $\mathcal{B}_V$ 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 $\{X_n\}_{n\in\mathbb{N}}$ to its invariant probability measure in operator norm on $\mathcal{B}_V$. 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. Advances in Applied Probability, 2014, 46 (4), pp.1036-1058. ⟨10.1239/aap/1418396242⟩. ⟨hal-00705523v4⟩