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.

翻译：我们针对一般可分离希尔伯特空间上的协方差算子（或等价地，中心化高斯测度）的Bures-Wasserstein空间建立了大数定律和中心极限定理。具体地，我们证明了在最小一阶矩条件下，以样本量为索引的经验重心序列几乎必然相对紧，其累积点构成总体重心。我们给出了极限唯一性的充分正则性条件。当极限唯一时，我们还在精细的矩条件和正则性条件下建立了中心极限定理。最后，我们证明了经验最优传输映射强算子收敛到其总体对应映射。尽管我们的结果自然扩展了有限维对应结果（包括相关的正则性条件），但由于一般设定中问题的函数性质，我们的技术方法截然不同。关键要素在于引出一类反映Bures-Wasserstein空间有序海涅-博雷尔性质的紧集。