Geodesic
On the convergence of the Metropolis-Hastings Markov chains
Serdica Mathematical Journal, Tome 43 (2017) no. 2, pp. 093-110.

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In this paper we study Markov chains associated with the Metropolis-Hastings algorithm. We consider 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 proposal density is enough for the chain to converge. The content of this work basically presents a stand alone proof that the reversibility along with the kernel positivity imply the convergence.
Keywords: Markov chain, Metropolis-Hastings algorithm, total variation distance, 60J05, 65C05, 60J22 
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     title = {On the convergence of the {Metropolis-Hastings} {Markov} chains},
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Tsvetkov, Dimiter; Hristov, Lyubomir; Angelova-Slavova, Ralitsa. On the convergence of the Metropolis-Hastings Markov chains. Serdica Mathematical Journal, Tome 43 (2017) no. 2, pp. 093-110. http://geodesic.mathdoc.fr/item/SMJ2_2017_43_2_a0/