Abstract

In this paper we use recent results of Jarner and Roberts (2000) to show polynomial convergence rates of MCMC algorithms with heavy tailed target distributions, in particular random walk Metropolis algorithms, Langevin algorithms and independence samplers. We also use similar methodology to consider polynomial convergence of the Gibbs sampler on a constrained state space. The main result for the random walk Metropolis algorithm is that heavy tailed proposal distributions lead to higher rates of convergence and thus to qualitatively better algorithms as measured for instance by the existence of central limit theorems for higher moments. Thus, the paper gives for the rst time a theoretical justi- cation for the common belief that heavy tailed proposal distributions improve convergence in the context of random walk Metropolis algorithms. Similar results are shown to hold for Langevin algorithms and the independence sampler, while results for the mixing of Gibbs samplers on uniform distr...

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