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We explicitly construct a coupling attaining Ornstein's $\bar{d}$-distancebetween ordered pairs of binary chains of infinite order. Our main tool is arepresentation of the transition probabilities of the coupled bivariate chainof infinite order as a countable mixture of Markov transition probabilities ofincreasing order. Under suitable conditions on the loss of memory of thechains, this representation implies that the coupled chain can be representedas a concatenation of iid sequence of bivariate finite random strings ofsymbols. The perfect simulation algorithm is based on the fact that we canidentify the first regeneration point to the left of the origin almost surely.

Perfect simulation of a coupling achieving the $\bar{d}$-distance between ordered pairs of binary chains of infinite order