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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.12853 |
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| _version_ | 1866910141783212032 |
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| author | Crane, Edward Holroyd, Alexander E. Russell, Erin |
| author_facet | Crane, Edward Holroyd, Alexander E. Russell, Erin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12853 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coupling Markov chains with a common image chain Crane, Edward Holroyd, Alexander E. Russell, Erin Probability 60J10 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. |
| title | Coupling Markov chains with a common image chain |
| topic | Probability 60J10 |
| url | https://arxiv.org/abs/2604.12853 |