Distance Covariance, Independence, and Pairwise Differences

Distance covariance (Székely et al. 2007) 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.