Open Access
September 2013 Distance covariance in metric spaces
Russell Lyons
Ann. Probab. 41(5): 3284-3305 (September 2013). DOI: 10.1214/12-AOP803


We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.


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Russell Lyons. "Distance covariance in metric spaces." Ann. Probab. 41 (5) 3284 - 3305, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1292.62087
MathSciNet: MR3127883
Digital Object Identifier: 10.1214/12-AOP803

Primary: 51K99 , 62G20 , 62H20
Secondary: 30L05 , 62H15

Keywords: Brownian covariance , Distance correlation , Hypothesis testing , independence , Negative type

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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