## The Annals of Probability

### Distance covariance in metric spaces

Russell Lyons

#### Abstract

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

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

Dates
First available in Project Euclid: 12 September 2013

https://projecteuclid.org/euclid.aop/1378991840

Digital Object Identifier
doi:10.1214/12-AOP803

Mathematical Reviews number (MathSciNet)
MR3127883

Zentralblatt MATH identifier
1292.62087

#### Citation

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

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