Positivity of hit-and-run and related algorithms

Daniel Rudolf, Mario Ullrich

Published 2012-12-18, updated 2013-11-12Version 2

We prove positivity of the Markov operators that correspond to the hit-and-run algorithm, random scan Gibbs sampler, slice sampler and an Metropolis algorithm with positive proposal. In all of these cases the positivity is independent of the state space and the stationary distribution. In particular, the results show that it is not necessary to consider the lazy versions of these Markov chains. The proof relies on a well known lemma which relates the positivity of the product M T M^*, for some operators M and T, to the positivity of T. It remains to find that kind of representation of the Markov operator with a positive operator T.