A diffusion process associated with Fr\'{e}chet means

TL;DR

This paper investigates the asymptotic behavior of sample Fréchet means on Riemannian manifolds, demonstrating that their scaled images converge to a diffusion process influenced by the manifold's geometry.

Contribution

It introduces a diffusion process limit for scaled Fréchet means on manifolds, extending Euclidean results to curved spaces with geometric dependence.

Findings

Weak convergence of scaled Fréchet means to a diffusion process

The limiting diffusion is a Brownian motion modulated by manifold geometry

Dependence on both covariance and global Riemannian structure

Abstract

This paper studies rescaled images, under $\exp^{-1}_{\mu}$, of the sample Fr\'{e}chet means of i.i.d. random variables $\{X_k\vert k\geq 1\}$ with Fr\'{e}chet mean $\mu$ on a Riemannian manifold. We show that, with appropriate scaling, these images converge weakly to a diffusion process. Similar to the Euclidean case, this limiting diffusion is a Brownian motion up to a linear transformation. However, in addition to the covariance structure of $\exp^{-1}_{\mu}(X_1)$, this linear transformation also depends on the global Riemannian structure of the manifold.