A coupling approach to Doob's theorem

TL;DR

This paper presents a coupling proof of Doob's theorem demonstrating convergence of transition probabilities in regular Markov processes with invariant measures, and extends results to cases with non-singular transition probabilities.

Contribution

Provides a new coupling proof of Doob's theorem and shows non-singularity of transition probabilities suffices for convergence for almost all initial conditions.

Findings

Coupling proof of Doob's theorem established.

Convergence of transition probabilities shown under non-singularity.

Results apply to a broader class of Markov processes.

Abstract

We provide a coupling proof of Doob's theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure $\mu$ converge to $\mu$ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for $\mu$-almost all initial conditions.