On maximal and distributional coupling (Q1083117)

The author proves the existence of a maximal coupling for discrete time stochastic sequences on a Polish state space. This result follows as a corollary to the result that there always exists a maximal distributional coupling for any two stochastic sequences defined on the same state space without any restrictions on the state space. The notion of distributional coupling was introduced by the author in Adv. Appl. Probab. 15, 531-561 (1983; Zbl 0512.60080) and reads as follows: A bivariate process \((\hat X,\hat Y)\) is a distributional coupling of the two processes X and Y if a) the marginal distributions of \((\hat X,\hat Y)\) are equal to the distributions of X and Y respectively and b) there exist two stochastic epochs T and T' such that the distribution of \(\hat X\) from time T on is equal to the distribution of \(\hat Y\) from time T' on.