Limit Theorems for Fr\'echet Mean Sets

TL;DR

This paper establishes limit theorems for Fréchet means in metric spaces, including strong laws, ergodic theorems, and large deviations, with new conditions for convergence in topologies like Fell.

Contribution

It provides the first sufficient conditions for strong law of large numbers for Fréchet mean sets in T2 topologies, extending classical results to set-valued random objects.

Findings

Limit theorems for Fréchet mean sets are derived from measure space results.

New sufficient conditions for strong law of large numbers in T2 topologies are established.

The results have implications for statistical analysis and computation involving Fréchet means.

Abstract

For $1\le p \le \infty$, the Fr\'echet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we prove a collection of limit theorems for Fr\'echet means and related objects, which, in general, constitute a sequence of random closed sets. On the one hand, we show that many limit theorems (a strong law of large numbers, an ergodic theorem, and a large deviations principle) can be simply descended from analogous theorems on the space of probability measures via purely topological considerations. On the other hand, we provide the first sufficient conditions for the strong law of large numbers to hold in a $T_2$ topology (in particular, the Fell topology), and we show that this condition is necessary in some special cases. We also discuss statistical and computational implications of the results herein.