Strong and Weak Laws of Large Numbers for Frechet Sample Means in Finite Metric Spaces

The Frechet mean 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 non-restricted Frechet sample means of all orders, thereby generalizing Ziezold's original result. In addition, we also show that both the restricted and non-restricted Frechet sample means are metric squared error (MSE) consistent. Convergence in probability and convergence in law of these sample estimators are also derived and the implications between these different modes of convergence are studied.