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Winter 2014 Hyperbolic space has strong negative type
Russell Lyons
Illinois J. Math. 58(4): 1009-1013 (Winter 2014). DOI: 10.1215/ijm/1446819297


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.


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Russell Lyons. "Hyperbolic space has strong negative type." Illinois J. Math. 58 (4) 1009 - 1013, Winter 2014.


Received: 4 October 2014; Revised: 2 March 2015; Published: Winter 2014
First available in Project Euclid: 6 November 2015

zbMATH: 1328.51005
MathSciNet: MR3421595
Digital Object Identifier: 10.1215/ijm/1446819297

Primary: 51K99 , 51M10
Secondary: 30L05 , 53C20

Rights: Copyright © 2014 University of Illinois at Urbana-Champaign


Vol.58 • No. 4 • Winter 2014
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