Distance covariance in metric spaces

Understand

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, 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.