Brazilian Journal of Probability and Statistics

Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables

Wilfrid S. Kendall and Huiling Le

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Abstract

We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fréchet means) of independent nonidentically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg–Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fréchet means, while extending its scope.

Article information

Source
Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 323-352.

Dates
First available in Project Euclid: 22 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1313973397

Digital Object Identifier
doi:10.1214/11-BJPS141

Mathematical Reviews number (MathSciNet)
MR2832889

Zentralblatt MATH identifier
1234.60025

Keywords
Central approximation theorem central limit theorem curvature empirical Fréchet mean exponential map Fréchet mean gradient Hessian Kähler manifold Lindeberg condition Newton’s method Riemannian centre of mass weak law of large numbers

Citation

Kendall, Wilfrid S.; Le, Huiling. Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables. Braz. J. Probab. Stat. 25 (2011), no. 3, 323--352. doi:10.1214/11-BJPS141. https://projecteuclid.org/euclid.bjps/1313973397


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