Distance Covariance, Independence, and Pairwise Differences

Distance covariance (Sz & eacute;kely, Rizzo, and Bakirov) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables X and Y. This approach deserves to be touched upon in modern courses on mathematical statistics. It makes use of distances of the type |X-X '| and |Y-Y '|, where (X ',Y ') is an independent copy of (X, Y). This raises natural questions about independence of variables like X-X ' and Y-Y ', about the connection between cov(|X-X '|,|Y-Y '|) and the covariance between doubly centered distances, and about necessary and sufficient conditions for independence. We show some basic results and present a new and nontechnical counterexample to a common fallacy, which provides more insight. We also show some motivating examples involving bivariate distributions and contingency tables, which can be used as didactic material for introducing distance correlation.