The Annals of Probability

Distance covariance in metric spaces

Russell Lyons

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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.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3284-3305.

First available in Project Euclid: 12 September 2013

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Zentralblatt MATH identifier

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62G20: Asymptotic properties 51K99: None of the above, but in this section
Secondary: 62H15: Hypothesis testing 30L05: Geometric embeddings of metric spaces

Negative type hypothesis testing independence distance correlation Brownian covariance


Lyons, Russell. Distance covariance in metric spaces. Ann. Probab. 41 (2013), no. 5, 3284--3305. doi:10.1214/12-AOP803.

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