A note on Metropolis-Hastings kernels for general state spaces (Q1296614)

Let \(P(x,dy)\) be a Markov transition kernel on a measurable space \(E\) with a specified invariant distribution \(\pi\). Consider the Hastings-Metropolis kernel \[ P(x,dy)= Q(x,dy)\alpha (x,y)+ \delta_x(dy) \int\bigl(1-\alpha (x,u)\bigr) Q(x,du), \] where \(Q(x,dy)\) is a transition kernel, \(\alpha (x,y): E\times E\to[0,1]\), and \(\delta_x\) is point mass at \(x\). First, necessary and sufficient conditions on \(Q\) and \(\alpha\) are given for the reversibility of this type of kernel. Next, in order to compare the performances of such kernels, a result of \textit{P. H. Peskun} [Biometrika 60, 607-612 (1973; Zbl 0271.62041)] on the ordering of asymptotic variances is extended from finite to general spaces.