$\alpha$-Congruence for Markov Processes

Abstract

We prove infinite-time extensions of invariance principles for certain random walks with essentially compact state spaces. The extensions are uniform-like in time since they use the $\bar{d}$-metric of the Bernoulli theory and imply the classical results. These are then generalized to couplings involving an isomorphism between the processes. In general a Doeblin-type condition is needed to hold for the walks but relaxation of this is indicated.