Strong Consistency of Set-Valued Fréchet Sample Means in Metric Spaces

The Fréchet mean generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, however, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For non-trivial examples, sequences of Fréchet sample mean sets may fail to converge in a set-analytical sense. Hence, it becomes necessary to consider other types of convergence. We show that a specific type of almost sure (a.s.) convergence for the Fréchet sample mean introduced by Ziezold (1977) is, in fact, equivalent to the consideration of the Kuratowski outer limit of a sequence of Fréchet sample means. Equipped with this outer limit, we prove different laws of large numbers for random variables taking values in separable (pseudo-)metric space with a bounded metric. In this setting, we describe strong laws of large numbers for the Fréchet sample mean. In particular, we demonstrate that all subsequences of Fréchet sample means converge to a subset of the theoretical mean. This result allows us to show that the Fréchet sample mean is metric squared error (MSE) consistent under the condition that their Kuratowski outer limits are non-empty. 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.