Fr\'echet Means in Infinite Dimensions

TL;DR

This paper extends the asymptotic theory of Fréchet means to infinite-dimensional metric spaces, establishing a continuity theorem that broadens applicability and reduces assumptions, with comprehensive example analysis.

Contribution

It proves a fundamental continuity theorem for Fréchet means in infinite-dimensional spaces, generalizing existing theory and removing prior limitations.

Findings

Unified asymptotic theory in infinite-dimensional spaces

Reduced moment assumptions to minimal levels

Removed the need for Fréchet mean uniqueness

Abstract

While there exists a well-developed asymptotic theory of Fr\'echet means of random variables taking values in a general "finite-dimensional" metric space, there are only a few known results in which the random variables can take values in an "infinite-dimensional" metric space. Presently, we fill this gap by proving a fundamental continuity theorem for Fr\'echet means in metric spaces which admit a suitably powerful notion of "weak convergence". This allows us to recover, strengthen, and generalize all known asymptotic theory for Fr\'echet means; in particular, we expand the possible geometric settings where such theory can be applied, we reduce the moment assumptions to the provably minimal possible, and we completely remove assumptions about uniqueness of the Fr\'echet mean. We also analyze many examples.