Strong Consistency of the Frechet Sample Mean in Bounded Metric Spaces

The Frechet mean or barycenter generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, however, the Frechet sample mean is not a consistent estimator of the theoretical Frechet mean. For non-trivial examples, the Frechet sample mean may fail to converge. Hence, it becomes necessary to consider other types of convergence. We show that a specific type of almost sure (a.s.) convergence for the Frechet sample mean introduced by Ziezold (1977) is, in fact, equivalent to the consideration of the Kuratowski outer limit of a sequence of Frechet sample means. Equipped with this outer limit, we prove different laws of large numbers for random variables taking values in a separable (pseudo-)metric space with a bounded metric. In this setting, we describe strong laws of large numbers for both the restricted and unrestricted Frechet sample means of all orders, thereby generalizing Ziezold's original result. In addition, we also show that both the restricted and unrestricted Frechet sample means are metric squared error (MSE) consistent. Interestingly, we derive a simple upper bound for this MSE, which is composed of the Frechet variance of the estimator and a bias term, thereby generalizing the classical decomposition of the mean squared error for estimators of real-valued random variables.