Central limit theorem for intrinsic Frechet means in smooth compact Riemannian manifolds

TL;DR

This paper establishes a central limit theorem for the Frechet mean on compact Riemannian manifolds, allowing for the cut locus to be within the support, and explicitly characterizes the non-standard term depending on the cut locus's co-dimension.

Contribution

It provides the first CLT for intrinsic means that includes cases where the cut locus is within the support, with explicit formulas for the non-standard term based on co-dimension.

Findings

CLT for intrinsic means with cut locus in support

Explicit expression for non-standard term when co-dimension is one

Allows cut locus of co-dimension one or two in the analysis

Abstract

We prove a central limit theorem (CLT) for the Frechet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Frechet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Frechet mean lies outside the support of the population distribution. So far as we are aware, the CLT in the present paper is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.