Distances between two-state Markov processes attainable by Markov joinings

Abstract:

The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the “Markov distance” between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the $\overline d$-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.