Strong laws of large numbers for generalizations of Fréchet mean sets

Abstract

A Fréchet mean of a random variable Y with values in a metric space (Q,d) is an element of the metric space that minimizes q↦Ed(Y,q)2. This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of Ed(Y,q)α for α>0, the minimizers of EH(d(Y,q)) for integrals H of non-decreasing functions, and the minimizers of Ec(Y,q) for a quite unrestricted class of cost functions 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.