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.
Highlights
By a classical result of [10] thedistance between stationary processes is identified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces
Thedistance extends Ornstein’s ddistance ([14]) and is applied to the information theoretic problem of source coding with a fidelity criterion, when the source statistics are incompletely known
To construct a class of optimal stationary couplings we define for a convex function f : Rn → R an equivariant map S : Ω → Ω
Summary
[10] introduced thedistance between two stationary probability measures μ, ν on EZ, where (E, ) is a separable, complete metric space (Polish space). To construct a class of optimal stationary couplings we define for a convex function f : Rn → R an equivariant map S : Ω → Ω. The following theorem implies that the class of equivariant maps defined in (2.6) gives a class of examples of optimal stationary couplings between stationary processes. Multivariate optimal coupling results as in Theorem 2.5 for the squared distance or later in Theorem 4.1 for general distance allow to compare higher dimensional marginals of two real stationary processes. For this purpose we consider a lifting of one-dimensional processes to multi-dimensional processes as follows. For general c not written as sum of one-dimensional cost functions the quantity c(sm)(μ, ν) has a meaning different from one-dimensional ones
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