A coupling approach to Doob's theorem

Alexei Kulik; Michael Scheutzow

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arXiv · 1407.8353

A coupling approach to Doob's theorem

Alexei Kulik·Michael Scheutzow

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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 $μ$ converge to $μ$ 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 $μ$-almost all initial conditions.

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Alexei Kulik, Michael Scheutzow. 2014-07-31. A coupling approach to Doob's theorem. https://arxiv.org/abs/1407.8353

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