Convergence of Markov chains in the relative supremum norm

Abstract

It is proved that the strong Doeblin condition (i.e., p^s(x,y) ≥ a_sπ(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - a_s)^1/s. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.