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On the measure of the cut locus of a Fréchet mean
Author(s) -
Le H.,
Barden D.
Publication year - 2014
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdu025
Subject(s) - mathematics , preprint , uniqueness , riemannian manifold , geodesic , locus (genetics) , independent and identically distributed random variables , measure (data warehouse) , central limit theorem , pure mathematics , maxima and minima , mathematical analysis , euclidean geometry , probability measure , manifold (fluid mechanics) , combinatorics , random variable , statistics , geometry , mechanical engineering , biochemistry , chemistry , database , world wide web , computer science , engineering , gene
One of the fundamental differences between the Central Limit Theorem for empirical Fréchet means obtained in [Kendall and Le, ‘Limit theorems for empirical Fréchet means of independent and non‐identically distributed manifold‐valued random variables’, Braz. J. Probab. Stat . 25 (2011) 323–352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Fréchet mean is zero. In [Hotz and Huckemann, ‘Intrinsic means on the circle: uniqueness, locus and asymptotics’, Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Fréchet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the p ‐energy function of a finite measure.