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

TL;DR

This paper establishes strong laws of large numbers for generalized Fréchet mean sets in metric spaces, covering various cost functions and minimal assumptions, extending classical results.

Contribution

It introduces new strong law results for generalized Fréchet means with diverse cost functions under minimal assumptions.

Findings

Convergence of empirical generalized Fréchet mean sets in outer limit.

Convergence in one-sided Hausdorff distance.

Applicable to a wide class of cost functions and metrics.

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.