  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 under a minimal first-moment condition, 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
elicitation of a class of compact sets that reflect an \emph{ordered}
Heine-Borel property of the Bures-Wasserstein space.
