Strong Consistency of Frechet Sample Mean Sets for Graph-Valued Random Variables

The Fréchet mean or barycenter generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For graph-valued random variables, for instance, the Fréchet sample mean may fail to converge to a unique value. Hence, it becomes necessary to consider the convergence of sequences of sets of graphs. We show that a specific type of almost sure (a.s.) convergence for the Fréchet sample mean previously introduced by Ziezold (1977) is, in fact, equivalent to the Kuratowski outer limit of a sequence of Fréchet sample means. Equipped with this outer limit, we provide a new proof of the strong consistency of the Fréchet sample mean for graph-valued random variables in separable (pseudo-)metric space. Our proof strategy exploits the fact that the metric of interest is bounded, since we are considering graphs over a finite number of vertices. In this setting, we describe two strong laws of large numbers for both the restricted and unrestricted Fréchet sample means of all orders, thereby generalizing a previous result, due to Sverdrup-Thygeson (1981).