On the convergence of the Metropolis-Hastings Markov chains

In this paper we consider Metropolis-Hastings Markov chains with absolutely continuous with respect to Lebesgue measure target and proposal distributions. We show that under some very general conditions the sequence of the powers of the conjugate transition operator has a strong limit in a properly defined Hilbert space described for example in Stroock [17]. Then we propose conditions under which the sequence of the successive densities of such a chain converges to the target density according to the total variation distance for any choice of the initial density. In particular we prove that the positiveness of the target and the proposal densities is enough for the chain to converge.