A diffusion process associated with Fréchet means

Abstract

This paper studies rescaled images, under $\exp^{-1}_{\mu}$, of the sample Fréchet means of i.i.d. random variables $\{X_{k}\vert k\geq 1\}$ with Fréchet 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.