Coupling Markov chains with a common image chain
Consider time-homogeneous discrete-time Markov chains $X$, $Y$, and $Z$ on countable state spaces, considered as stochastic processes with specified initial distributions. Suppose for maps $f$ and $g$ that $(f(X_t))_{t \ge 0}$ and $(g(Y_t))_{t \ge 0}$ are both equal in law to $Z$. We prove that $X$ and $Y$ can be coupled so that $(X_t, Y_t)_{t \ge 0}$ is a homogeneous Markov chain with $f(X_t) = g(Y_t)$ for all $t \ge 0$. Without the assumption that $Z$ is Markov, no such Markov coupling exists in general, even an inhomogeneous one. Moreover, we give an explicit construction of such a coupling, with the additional property that $X$ and $Y$ are conditionally independent given the entire trajectory $(f(X_t))_{t \ge 0}$. Under the further assumption that $X$ and $Y$ are stationary, we construct a coupling having the above properties that is also stationary. In this case, conditional independence holds for the corresponding two-sided chains indexed by $\mathbb{Z}$ (but not necessarily for the one-sided versions). We prove further properties of our couplings in special cases where $f$ or $g$ satisfies the strong lumping condition (also known as Dynkin's condition) or the exact lumping condition (also known as the Pitman-Rogers condition). When $f$ is a strong lumping and $g$ is an exact lumping, we show that our coupling coincides with an intertwining of Markov chains as constructed by Diaconis and Fill.