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.
Categories: math.PR