A diffusion process associated with Fréchet means

Le, Huiling

Abstract

This paper studies rescaled images, under exp−1μ, of the sample Fréchet means of i.i.d. random variables {Xk|k≥1} with Fréchet mean μ on a Rie-mannian 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−1μ(X1), this linear transformation also depends on the global Riemannian structure of the manifold

Citation

Le, H. (2015). A diffusion process associated with Fréchet means. Annals of Applied Probability, 25(6), https://doi.org/10.1214/14-AAP1066

• Journal Article Type: Article
• Acceptance Date: Sep 9, 2014
• Publication Date: Jan 1, 2015
• Deposit Date: Apr 11, 2016
• Publicly Available Date: Apr 11, 2016
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Electronic ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics (IMS)
• Peer Reviewed: Peer Reviewed
• Volume: 25
• Issue: 6
• DOI: https://doi.org/10.1214/14-AAP1066
• Keywords: limiting diffusion; rescaled Frechet means; weak convergence
• Public URL: https://nottingham-repository.worktribe.com/output/988633
• Publisher URL: http://projecteuclid.org/euclid.aoap/1443703768