Equivalence of two orthogonalities between probability measures

TL;DR

This paper establishes a geometric equivalence between two notions of orthogonality for probability measures with mean zero and finite variance in Euclidean space, linking measure orthogonality to the orthogonality of their support spaces.

Contribution

It proves that Wasserstein orthogonality of probability measures is equivalent to the orthogonality of their support spaces, providing a new geometric insight.

Findings

Wasserstein orthogonality implies support space orthogonality.

Support space orthogonality implies Wasserstein orthogonality.

The equivalence holds for measures with mean zero and finite variance.

Abstract

Given any two probability measures on a Euclidean space with mean 0 and finite variance, we demonstrate that the two probability measures are orthogonal in the sense of Wasserstein geometry if and only if the two spaces by spanned by the supports of each probability measure are orthogonal.