Zero-couplings of infinite measures with cyclically monotone support and multivariate regular variation

TL;DR

This paper develops a theory of cyclically monotone transport plans between infinite measures in multivariate regular variation, establishing existence, uniqueness, and representation results, and applying them to tail limit analysis.

Contribution

It introduces the concept of zero-couplings for infinite measures, proves their existence and uniqueness under certain conditions, and connects these to tail limits in multivariate regular variation.

Findings

Existence of cyclically monotone zero-couplings for arbitrary measure pairs.

Uniqueness of zero-couplings under Hausdorff-dimension and infinite mass conditions.

Representation of couplings via gradients of convex functions.

Abstract

We study cyclically monotone transport plans between measures in $\mathrm{M}_0(\mathbb{R}^d)$, the class of Borel measures on $\mathbb{R}^d \setminus \{0\}$ that are finite on sets bounded away from the origin but may have infinite total mass. We avoid moment assumptions and allow the transport cost to be infinite. This framework naturally arises for exponent measures in multivariate regular variation and includes other examples such as L\'evy measures. We introduce the notion of a zero-coupling and establish existence of cyclically monotone zero-couplings for arbitrary pairs of measures in $\mathrm{M}_0(\mathbb{R}^d)$. Under a Hausdorff-dimension condition on the first measure and when at least one of the two measures has infinite mass, we prove uniqueness of the cyclically monotone zero-coupling, yielding an analogue of the Brenier--McCann theorem in this infinite-measure setting. We further derive a representation of such couplings through gradients of closed convex functions and identify conditions under which the zero-coupling is proper in the sense that the second measure is equal to the restriction to the punctured space of the push-forward of the first measure by a cyclically monotone transport map. Finally, we apply these results to regularly varying probability measures. We show that a cyclically monotone coupling between two such distributions admits a tail limit that coincides with the unique proper cyclically monotone zero-coupling between the corresponding exponent measures.