Optimal scaling of the Random Walk Metropolis algorithm under Lp mean   differentiability
  This paper considers the optimal scaling problem for high-dimensional random walk Metropolis algorithms for densities which are differentiable in Lp mean but which may be irregular at some points (like the Laplace density for example) and/or are supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to infinity. Because the log-density might be non-differentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of [6]. This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors. 
Alain Durmus (LTCI), Sylvain Le Corff, Eric Moulines (CMAP), Gareth O.   Roberts
math.PR math.ST stat.TH