Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance

We consider a Metropolis–Hastings method with proposal N(x,hG(x)?1), where x is the current state, and study its ergodicity properties. show that suitable choices of G(x) can change these properties compared to Random Walk Metropolis case N(x,h?), either for better or worse. find if variance allowed grow unboundedly in tails distribution then geometric be established when target algorithm has are heavier than exponential, contrast case, but growth rate must carefully controlled prevent rejection approaching unity. also illustrate judicious choice result geometrically ergodic chain probability concentrates on an ever narrower ridge tails, something again not true Metropolis.