Strong laws of large numbers for Fréchet means

12/23/2020

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by   Steven N. Evans, et al.

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For 1 ≤ p < ∞, the Fréchet p-mean of a probability distribution μ on a metric space (X,d) is the set F_p(μ) := min_x∈ X∫_Xd^p(x,y) dμ(y), which is taken to be empty if no minimizer exists. Given a sequence (Y_i)_i ∈ℕ of independent, identically distributed random samples from some probability measure μ on X, the Fréchet p-means of the empirical measures, F_p(1/n∑_i=1^nδ_Y_i) form a sequence of random closed subsets of X. We investigate the senses in which this sequence of random closed sets and related objects converge almost surely as n →∞.