Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets

12/23/2020

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by   Christof Schötz, et al.

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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 ↦𝔼[d(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 𝔼[d(Y, q)^α] for α > 0, the minimizers of 𝔼[H(d(Y, q))] for integrals H of non-decreasing functions, and the minimizers of 𝔼[𝔠(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.