We establish strong laws of large numbers and central limit theorems in the Bures-Wasserstein space of the covariance operator over a general separable Hilbert space, or the equivalent central Gaussian measure. Specifically, under minimal first-moment conditions, the empirical centroid sequence indexed by sample size is almost certainly relatively compact, indicating that the accumulated points constitute the population centroids. indicates Give a sufficient regularity condition for the limit to be unique. If the limit is unique, the central limit theorem is also established under a sophisticated pair of moment and regularity conditions. Finally, we prove strong operator convergence to the corresponding population of empirical optimal transport maps. Our results naturally extend the finite-dimensional counterpart to include relevant regularity conditions, but the functional nature of the problem in the general setting makes our approach distinctly different. The key element is to derive a class of compact sets that reflect the \emph{ordered} Heine-Borel property of the Bures-Wasserstein space.
