A diffusion process associated with Fréchet means

Abstract

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

Published at http://dx.doi.org/10.1214/14-AAP1066 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Large sample theory of intrinsic and extrinsic sample means on manifolds--II