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

• 成果类型: Article
• 署名作者: Hotz, Thomas; Le, Huiling; Wood, Andrew T. A.
• 署名单位: Technische Universitat Ilmenau; University of Nottingham; Australian National University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01291-3
• 发表日期: 2024
• 页码: 1219-1246
• 关键词: extrinsic sample means; cut locus

摘要

We prove a central limit theorem (CLT) for the Fr & eacute;chet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fr & eacute;chet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fr & eacute;chet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fr & eacute;chet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon-Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here 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.