A note on diffusive/random-walk behaviour in Metropolis--Hastings algorithms

We prove a general result that if a Metropolis--Hastings algorithm has a proposal that is not geometrically ergodic and the acceptance rate approaches unity at a suitable rate as the state variable becomes large, then the Metropolised chain will also not be geometrically ergodic. Our conditions seem stronger than might be expected, but are shown to be necessary through a counterexample. We then turn our attention to the random walk and guided walk Metropolis algorithms. We show that if the target distribution has polynomial tails the latter converges at twice the polynomial rate of the former, but that if instead the target distribution has strictly convex potential then the random walk Metropolis behaves as a $1/2$-lazy version of the guided walk Metropolis when the state variable is large, and therefore moves at a similar (ballistic) speed.