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

This paper studies rescaled images, under ${\exp }_{\mu }^{-1}$, of the sample Fr'{e}chet means of i.i.d. random variables $\{X_k\mathrm{\mid }k\ge 1\}$ 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 ${\exp }_{\mu }^{-1}(X_1)$, this linear transformation also depends on the global Riemannian structure of the manifold.