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

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 $r\mathrm{\_}ess(P)$ of $P$. In this paper a computable bound of $r\mathrm{\_}ess(P)$ 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 : $\mathrm{\mid }u\mathrm{\mid }\text{textgreater}s\Rightarrow q(x,x+u)=0$ (for some $s\text{textgreater}0$). This result is illustrated with Random Walk Metropolis-Hastings kernels.