Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds

TL;DR

This paper introduces Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to Fréchet counterparts, establishing laws of large numbers and central limit theorems without geometric restrictions.

Contribution

It develops a novel framework for approximating Fréchet means on manifolds using Varadhan functions, with new laws of large numbers and CLTs that bypass cut locus assumptions.

Findings

Proves uniform laws of large numbers for empirical Varadhan functions, variances, and means.

Establishes CLTs for Varadhan functions and variances at fixed times.

Links small time asymptotics to CLT for Fréchet means without geometric restrictions.

Abstract

Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit theorems for Varadhan functions and variances for each fixed $t\ge0$, and for Varadhan means for each fixed $t>0$. By studying small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fr\'echet means, without assumptions on the geometry of the cut locus.