Strong Laws of Large Numbers for Generalizations of Fr\'echet Mean Sets

Abstract

A Fr\'echet mean of a random variable $Y$ with values in a metric space $(\mathcal Q, d)$ is an element of the metric space that minimizes $q \mapsto \mathbb E[d(Y,q)^2]$. This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fr\'echet means. Following generalizations are considered: the minimizers of $\mathbb E[d(Y, q)^\alpha]$ for $\alpha > 0$, the minimizers of $\mathbb E[H(d(Y, q))]$ for integrals $H$ of non-decreasing functions, and the minimizers of $\mathbb E[\mathfrak c(Y, q)]$ for a quite unrestricted class of cost functions $\mathfrak c$. We show convergence of empirical versions of these sets in outer limit and in one-sided Hausdorff distance. The derived results require only minimal assumptions.