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

Let ππ be a positive continuous target density on RR. Let PP be the Metropolis–Hastings operator on the Lebesgue space L2(π)L2(π) corresponding to a proposal Markov kernel QQ on RR. When using the quasi-compactness method to estimate the spectral gap of PP, a mandatory first step is to obtain an accurate bound of the essential spectral radius ress(P)ress(P) of PP. In this paper a computable bound of ress(P)ress(P) is obtained under the following assumption on the proposal kernel: QQ has a bounded continuous density q(x,y)q(x,y) on R2R2 satisfying the following finite range assumption : |u|>s⇒q(x,x+u)=0 (for some s>0s>0). This result is illustrated with Random Walk Metropolis–Hastings kernels.