A computable bound of the essential spectral radius of finite range Metropolis–Hastings kernels

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 $\pi$ be a positive continuous target density on $\mathbb{R}$. Let $P$ be the Metropolis-Hastings operator on the Lebesgue space $\mathbb{L}^2(\pi)$ corresponding to a proposal Markov kernel $Q$ on $\mathbb{R}$. When using the quasi-compactness method to estimate the spectral gap of $P$, a mandatory first step is to obtain an accurate bound of the essential spectral radius $r_{ess}(P)$ of $P$. In this paper a computable bound of $r_{ess}(P)$ is obtained under the following assumption on the proposal kernel: $Q$ has a bounded continuous density $q(x,y)$ on $\mathbb{R}^2$ satisfying the following finite range assumption : $|u| > s \, \Rightarrow\, q(x,x+u) = 0$ (for some $s>0$). This result is illustrated with Random Walk Metropolis-Hastings kernels.

Mots clés: Markov chain operator; Metropolis–Hastings algorithms; Spectral gap

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. A computable bound of the essential spectral radius of finite range Metropolis–Hastings kernels. Statistics and Probability Letters, 2016, 117, pp.72-79. ⟨10.1016/j.spl.2016.05.007⟩. ⟨hal-01356804⟩