Erin Russell

Coupling Markov chains with Markov chains

Consider discrete-time countable-state Markov chains X, Y and Z (with initial distributions specified). Suppose for maps f and g that (f (X_t))_{t ≥ 0} and (g(Y_t))_{t ≥ 0} are both equal in law to Z. With the help of an example concerning forest fires, we will discuss results regarding the coupling of X and Y so that (X_t , Y_t )_{t ≥ 0} is a Markov chain with f(X_t) = g(Y_t) for all t ≥ 0. Without the assumption that Z is Markov, no such Markov coupling exists in general. We will then investigate two important special cases: when f is a "strong lumping" (related to Dynkin's condition) and when f is an "exact lumping" (related to the Pitman--Rogers condition). We will relate our findings to a concept known as "intertwining of Markov chains", first seen in Diaconis and Fill (1990).