A note on faithful coupling of Markov chains

One often needs to turn a coupling (Xi, Yi)i≥0 of a Markov chain into a sticky coupling where once XT = YT at some T , then from then on, at each subsequent time step T ′ ≥ T , we shall have XT ′ = YT ′ . However, not all of what are considered couplings in literature, even Markovian couplings, can be turned into sticky couplings, as proved by Rosenthal [Ro97] through a counter example. Rosenthal then proposed a strengthening of the Markovian coupling notion, termed as faithful coupling, from which a sticky coupling can indeed be obtained. We identify the reason why a sticky coupling could not be obtained in the counter example of Rosenthal, which motivates us to define a type of coupling which can obviously be turned into a sticky coupling. We show then that the new type of coupling that we define, and the faithful coupling as defined by Rosenthal [Ro97], are actually identical. Our note may be seen as a demonstration of the naturalness of the notion of faithful coupling.