Coupling Markov chains with a common image chain

TL;DR

This paper constructs explicit Markov couplings for chains with shared image processes, exploring conditions for stationarity, independence, and connections to lumping conditions and intertwining of Markov chains.

Contribution

It provides a novel explicit construction of Markov couplings with common image chains, including stationary and conditionally independent variants, under various lumping conditions.

Findings

Explicit coupling construction with shared image process

Conditional independence given the entire trajectory of the image process

Connections to lumping conditions and intertwining of Markov chains

Abstract

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.