Gaussian Fluctuations of the Gradient of the Fréchet Function and Nonstandard Asymptotics of Sample Fréchet Means

Abstract

We establish a central limit theorem for the gradient of the population Fréchet function in local Euclidean coordinates on a metric space. Assuming a unique population mean, a consistent measurable sequence of sample means, semiconcavity and moment bounds for the squared distance functions, and local $C^{1,α}$ regularity of the population Fréchet function, we prove an asymptotic linear representation for the gradient of the population Fréchet function evaluated at the sample mean. The leading term is the negative empirical average of the gradients of the squared distance functions at the population mean; these gradients exist almost surely under our assumptions. On connected complete Riemannian manifolds, uniqueness of the population mean, existence of a $(2+η)$-moment, and local $C^{1,α}$ regularity suffice, and the leading term is expressed through the logarithm map. To control the empirical process uniformly over a neighborhood where squared distance functions may be nonsmooth, we use one-sided partial derivatives of the squared distance functions. The key estimate is a polynomial bound on the $L^2$ bracketing numbers of the resulting function classes. We also give a sufficient condition for the regularity of the population Fréchet function through probability bounds for neighborhoods of cut loci. We then show how the local behavior of the population Fréchet function transforms these Gaussian fluctuations into limit distributions for suitably rescaled sample Fréchet means, allowing for anisotropic scaling and non-Gaussian limits. Polynomial, logarithmic, and oscillatory examples illustrate the applicability and limitations of this approach.