The Annals of Probability
- Ann. Probab.
- Volume 41, Number 5 (2013), 3284-3305.
Distance covariance in metric spaces
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.
Ann. Probab., Volume 41, Number 5 (2013), 3284-3305.
First available in Project Euclid: 12 September 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
Lyons, Russell. Distance covariance in metric spaces. Ann. Probab. 41 (2013), no. 5, 3284--3305. doi:10.1214/12-AOP803. https://projecteuclid.org/euclid.aop/1378991840