Optimal scaling for the transient phase of Metropolis Hastings algorithms: the longtime behavior.

Summary We consider the Random Walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known (see Roberts et al. (1997))) that, in the limit $n \to \infty$, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain.