Hyperbolic Space Has Strong Negative Type

TL;DR

This paper proves that hyperbolic spaces possess strong negative type, which ensures the uniqueness of probability measures based on expected distances and confirms the consistency of the distance covariance test for independence in these spaces.

Contribution

It establishes that hyperbolic spaces have strong negative type, extending known properties and implications for probability measures and statistical independence testing.

Findings

Hyperbolic spaces have strong negative type.

Expected distances uniquely determine probability measures.

Distance covariance test is consistent in hyperbolic spaces.

Abstract

It is known that hyperbolic spaces have strict negative type, a condition on the distances of any finite subset of points. We show that they have strong negative type, a condition on every probability distribution of points (with integrable distance to a fixed point). This implies that the function of expected distances to points determines the probability measure uniquely. It also implies that the distance covariance test for stochastic independence, introduced by Sz\'ekely, Rizzo and Bakirov, is consistent against all alternatives in hyperbolic spaces. We prove this by showing an analogue of the Cram\'er-Wold device.