The Annals of Applied Probability

A diffusion process associated with Fréchet means

Huiling Le

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3033-3046.

Dates
Received: July 2013
Revised: September 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703768

Digital Object Identifier
doi:10.1214/14-AAP1066

Mathematical Reviews number (MathSciNet)
MR3404630

Zentralblatt MATH identifier
1328.58035

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems

Keywords
Limiting diffusion rescaled Fréchet means weak convergence

Citation

Le, Huiling. A diffusion process associated with Fréchet means. Ann. Appl. Probab. 25 (2015), no. 6, 3033--3046. doi:10.1214/14-AAP1066. https://projecteuclid.org/euclid.aoap/1443703768


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