On optimal stationary couplings between stationary processes

TL;DR

This paper explores optimal stationary couplings between stationary processes, extending classical results to random fields and nonmetric distances, and provides explicit examples and construction methods under certain geometric conditions.

Contribution

It introduces new classes of optimal stationary couplings, extends the $ar o$ distance to random fields and nonmetric distances, and offers a construction method under positive curvature assumptions.

Findings

Derived explicit formulas for $ar o$ distance in specific cases

Extended the $ar o$ distance to random fields and nonmetric distances

Provided a construction method for optimal stationary $ar c$-couplings

Abstract

By a classical result of Gray, Neuhoff and Shields (1975) the $\bar\varrho$ distance between stationary processes is identified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces. This is a modification of the optimal coupling problem from Monge--Kantorovich theory. In this paper we derive some general classes of examples of optimal stationary couplings which allow to calculate the $\bar\varrho$ distance in these cases in explicit form. We also extend the $\bar\varrho$ distance to random fields and to general nonmetric distance functions and give a construction method for optimal stationary $\bar c$-couplings. Our assumptions need in this case a geometric positive curvature condition.