  This paper studies the mixing time of certain adaptive Markov Chain Monte
Carlo algorithms. Under some regularity conditions, we show that the
convergence rate of Importance Resampling MCMC (IRMCMC) algorithm, measured in
terms of the total variation distance is $O(n^{-1})$, and by means of an
example, we establish that in general, this algorithm does not converge at a
faster rate. We also study the Equi-Energy sampler and establish that its
mixing time is of order $O(n^{-1/2})$.
