On the Poisson equation for Metropolis-Hastings chains

This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis-Hastings chain $\mathrm{\Phi }$. The approximations give rise to a natural sequence of control variates for the ergodic average $S_k(F)=(1\mathrm{/}k){\sum }_{i=1}^{k}F({\mathrm{\Phi }}_i)$, where $F$ is the force function in the Poisson equation. The main result of the paper shows that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and gives a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.