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

TL;DR

This paper presents an explicit construction of a coupling that achieves Ornstein's ar{d}-distance between ordered pairs of binary chains of infinite order, using a mixture of Markov transition probabilities and a perfect simulation algorithm.

Contribution

It introduces a novel coupling construction for infinite-order binary chains that attains the ar{d}-distance and provides a perfect simulation method based on regeneration points.

Findings

Coupling attains Ornstein's ar{d}-distance for infinite binary chains.

Representation as a mixture of Markov transition probabilities.

Perfect simulation algorithm with almost sure regeneration point identification.

Abstract

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