Extending Characterizations of Multivariate Laws via Distance Distributions

Abstract: We extend a theorem of Maa, Pearl, and Bartoszynski, which links equality of interpoint distance distributions to equality of underlying multivariate distributions, beyond the restrictive class of homogeneous, translation-invariant distance functions. Our approach replaces geometric assumptions on the distance with analytic conditions: volume-regularity of distance-induced balls, Lebesgue differentiability with respect to the distance, and bounded centered oscillations of densities. Under these conditions, equality of interpoint distance distributions continues to imply equality of the generating laws. The result persists under monotone continuous transformations of homogeneous, translation-invariant distances, recovering the original statement, and it extends to compact Riemannian manifolds equipped with the geodesic metric. We further develop a quantitative version of the theorem, i.e., inequalities that connect discrepancies of interpoint distance distributions to the $L^2$-distance between densities, and obtain explicit rates under Ahlfors $\alpha$-regularity of the distance function and $\beta$-H\"older continuity of densities, capturing dependence on dimensionality. Several representative examples illustrate the applicability of the generalization to domain-specific distances used in modern statistics. The examples include non-homogeneous non-translation invariant distances such as Canberra, entropic distances, and the Bray--Curtis dissimilarity.