Logarithmic energy distances and Gini covariance for Hilbert-valued random elements

Abstract

For $\alpha\in(0,2)$, the generalized energy distance and the Gini covariance statistic are based on kernels of the form $(x,y)\mapsto \|x-y\|^\alpha$, where $\|\cdot\|$ denotes the norm in a real separable Hilbert space. This paper investigates the boundary regime $\alpha\downarrow 0$. After suitable normalization, the corresponding energy distance converges to a logarithmic energy distance involving the kernel $(x,y)\mapsto\log\|x-y\|$. We establish that the resulting logarithmic energy distance retains the fundamental characterization property of ordinary energy distances in separable Hilbert spaces and derive a representation in terms of Gaussian-kernel maximum mean discrepancies. Motivated by this representation, we introduce a logarithmic Gini covariance for the $k$-sample problem and investigate its structural and asymptotic properties. In particular, we derive a representation in terms of pairwise logarithmic energy distances, establish a characterization theorem for equality of distributions, develop asymptotic null and alternative theory for the corresponding empirical statistic, and discuss permutation-based implementation. The logarithmic framework reveals a new boundary phenomenon within the family of energy-type statistics and provides connections with kernel methods, functional data analysis, and high-dimensional inference.