A note on faithful coupling of Markov chains

TL;DR

This paper investigates the conditions under which Markov chain couplings can be transformed into sticky couplings, clarifies the concept of faithful coupling, and proves its equivalence to a newly defined coupling type, demonstrating its naturalness.

Contribution

The paper shows that faithful coupling and a newly defined coupling type are equivalent, clarifying the conditions for turning couplings into sticky couplings.

Findings

Faithful coupling is equivalent to a new, naturally defined coupling type.

Not all Markovian couplings can be turned into sticky couplings, as shown by counterexamples.

The paper clarifies the naturalness and conditions of faithful coupling in Markov chains.

Abstract

One often needs to turn a coupling $(X_i, Y_i)_{i\geq 0}$ of a Markov chain into a sticky coupling where once $X_T = Y_T$ at some $T$, then from then on, at each subsequent time step $T'\geq T$, we shall have $X_{T'} = Y_{T'}$. However, not all of what are considered couplings in literature, even Markovian couplings, can be turned into sticky couplings, as proved by Rosenthal 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, are actually identical. Our note may be seen as a demonstration of the naturalness of the notion of faithful coupling.