  In a recent paper `The equi-energy sampler with applications statistical
inference and statistical mechanics' [Ann. Stat. 34 (2006) 1581--1619], Kou,
Zhou & Wong have presented a new stochastic simulation method called the
equi-energy (EE) sampler. This technique is designed to simulate from a
probability measure $\pi$, 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 Atchad\'e & Liu (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 provide a new proof of convergence of the
equi-energy sampler based on the Poisson equation and on the theory developed
in Andrieu et al. (2007) for \emph{Non-Linear} Markov chain Monte Carlo (MCMC).
The objective of this note is to provide a proof of correctness of the EE
sampler when there is only one feeding chain; the general case requires a much
more technical approach than is suitable for a short note. In addition, we also
seek to highlight the difficulties associated with the analysis of this type of
algorithm and present the main techniques that may be adopted to prove the
convergence of it.
