# Distance covariance in metric spaces

```@article{Lyons2013DistanceCI,
title={Distance covariance in metric spaces},
author={Russell Lyons},
journal={Annals of Probability},
year={2013},
volume={41},
pages={3284-3305}
}```
• R. Lyons
• Published 28 June 2011
• Mathematics
• Annals of Probability
We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Szekely, 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… Expand
112 Citations
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#### References

SHOWING 1-10 OF 48 REFERENCES
On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space
1. W. A. Wilson ([9])2 has recently investigated those metric spaces which arise from a metric space by taking as its new metric a suitable (one variable) function of the old one. He considered inExpand
On potentials of measures in Banach spaces
• Mathematics
• 1987
in M with respect to the Kobayashi metric, such that the closure of V in M is compact (the existence of such a ball follows from the condition that M is hyperbolic), U be a ball of the same radius inExpand
Metric spaces and positive definite functions
As poo we get the space Em with the distance function maxi-, ... I xi X. Let, furthermore, lP stand for the space of real sequences with the series of pth powers of the absolute values convergent.Expand
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic ofExpand
Isometric operators in vector-valued LP-spaces
The note is devoted to the generalization of A. I. Plotkin's theorem on the equimeasurability of a function and of its image under a LP-isometry to the case of isometric operators, acting in certainExpand
L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry
• A. Naor
• Mathematics, Computer Science
• ArXiv
• 2010
This work surveys connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization and explains how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. Expand
DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I
• Mathematics
• 2009
Let ( X , d ) be a compact metric space and let ℳ( X ) denote the space of all finite signed Borel measures on  X . Define I :ℳ( X )→ℝ by and set M ( X )=sup I ( μ ), where μ ranges over theExpand
Brownian distance covariance
• Mathematics
• 2009
Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlationExpand
Convergence of probability measures
The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates theExpand
Measuring and testing dependence by correlation of distances
• Mathematics
• 2007
Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike theExpand