Convergence of Markov chains in the relative supremum norm (Q2725301)

This paper shows that the strong Doeblin condition implies geometric convergence in the relative supremum norm (i.e. \(\sup_x\{(p(x)- \pi(x))/\pi(x)\})\) if the initial relative error is bounded. The strong Doeblin condition is that the \(s\)-step transition density \(p^s(x,y)\) satisfies \(p^s(x,y)\geq a_s\pi(y)\) for all \(x\), \(y\) in the state space. The convergence rate is \((1- a_s)^{1/s}\). For Markov chains satisfying the detailed balance condition and weak continuity condition, the strong Doeblin condition is equivalent to convergence in the relative supremum norm.