Large Sample Theory for Bures-Wasserstein Barycentres

TL;DR

This paper develops large sample statistical theory for Bures-Wasserstein barycentres in infinite-dimensional Hilbert spaces, including laws of large numbers and central limit theorems, with applications to Gaussian measures and covariance operators.

Contribution

It extends finite-dimensional Bures-Wasserstein barycentre theory to infinite-dimensional spaces, establishing strong laws, CLTs, and operator convergence results with new techniques.

Findings

Empirical barycentres are relatively compact and converge to population barycentres.

Under regularity conditions, the limit barycentre is unique.

Empirical optimal transport maps converge strongly to population maps.

Abstract

We establish a strong law of large numbers and a central limit theorem in the Bures-Wasserstein space of covariance operators -- or equivalently centred Gaussian measures -- over a general separable Hilbert space. Specifically, we show that empirical barycentre sequences indexed by sample size are almost certainly relatively compact, with accumulation points comprising population barycentres. We give a sufficient regularity condition for the limit to be unique. When the limit is unique, we also establish a central limit theorem under a refined pair of moment and regularity conditions. Finally, we prove strong operator convergence of the empirical optimal transport maps to their population counterparts. Though our results naturally extend finite-dimensional counterparts, including associated regularity conditions, our techniques are distinctly different owing to the functional nature of the problem in the general setting. A key element is the characterisation of compact sets in the Bures-Wasserstein topology that reflects an ordered Heine-Borel property of the Bures-Wasserstein space.