Convergence of the equi-energy sampler

Abstract
In a recent paper 'The equi-energy sampler with applications statistical inference and statistical mechanics' Kou, Zhou & Wong (Ann. Statist., 34, 1581-1619, 2006.) have presented a new stochastic simulation method called the equi-energy (EE) sampler. This technique is designed to simulate from a probability measure , perhaps only known up to a normalizing constant. The authors demonstrate that the sampler performs well in quite challenging problems but their convergence results (Theorem 2) appear incomplete. This was pointed out, in the discussion of the paper, by Atchadé & Liu (Ann. Statist., 34, 1620-1628, 2006.) who proposed an alternative convergence proof. However, this alternative proof, whilst theoretically correct, does not correspond to the algorithm that is implemented. In this note we consider the difficulties of the proofs as well as pointing out an alternative convergence result established by the authors (Andrieu et al. 2007b).