Abstract
We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.
Highlights
We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of couplings
We provide an example showing that uniqueness can fail if the state space is separable and metric even though a coupling defined via a continuous and positive definite function exists
One important question in the theory of Markov processes is that of existence and uniqueness of invariant probability measures
Summary
One important question in the theory of Markov processes is that of existence and uniqueness of invariant probability measures (ipms). No matter how we couple the chains starting in A and in B: most of the time, the first process is in A and the second one is in B and so their distance is at least equal to the distance of the sets A and B (which is strictly positive), contradicting the usual coupling assumption that there exists a coupling for which the processes starting in A and in B are very close for large times. This argument still holds if couplings are replaced by generalized couplings (see the definition below). We present and prove the main result along the lines [5] and [7] but using these propositions instead of ergodicity and inner regularity in the Polish case
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