## Illinois Journal of Mathematics

### Hyperbolic space has strong negative type

Russell Lyons

#### 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 Székely, Rizzo and Bakirov, is consistent against all alternatives in hyperbolic spaces. We prove this by showing an analogue of the Cramér–Wold device.

#### Article information

Source
Illinois J. Math., Volume 58, Number 4 (2014), 1009-1013.

Dates
Revised: 2 March 2015
First available in Project Euclid: 6 November 2015

https://projecteuclid.org/euclid.ijm/1446819297

Digital Object Identifier
doi:10.1215/ijm/1446819297

Mathematical Reviews number (MathSciNet)
MR3421595

Zentralblatt MATH identifier
1328.51005

#### Citation

Lyons, Russell. Hyperbolic space has strong negative type. Illinois J. Math. 58 (2014), no. 4, 1009--1013. doi:10.1215/ijm/1446819297. https://projecteuclid.org/euclid.ijm/1446819297

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