Uniform Limit Theorems for Diffusion Means on Compact Riemannian Manifolds

Abstract

Motivated by Varadhan's theorem, we study Varadhan functions and variances on compact Riemannian manifolds together with their minimizers, the diffusion means introduced in Eltzner et al. (2023). Given independent and identically distributed samples, we prove uniform laws of large numbers and uniform central limit theorems for their empirical versions. Through the small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fréchet means on compact Riemannian manifolds by recovering the ``non-standard'' correction term from Hotz et al. (2024).