Coupling methods are used to obtain a structure theorem for the atomic decomposition of the tail $\sigma$-algebra of an arbitrary nonhomogeneous Markov chain. Various related results are also derived by coupling.
"Partial Coupling and Loss of Memory for Markov Chains." Ann. Probab. 4 (5) 850 - 858, October, 1976. https://doi.org/10.1214/aop/1176995989