On optimal stationary couplings between stationary processes

Abstract

By a classical result of Gray, Neuhoff and Shields (1975) the rhobar-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 rhobar distance in these cases in explicit form. We also extend the rhobar-distance to random fields and to general nonmetric distance functions and give a construction method for optimal stationary cbar-couplings. Our assumptions need in this case a geometric positive curvature condition.