  Sz\'{e}kely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769-2794) and
Sz\'{e}kely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236-1265), in two seminal
papers, introduced the powerful concept of distance correlation as a measure of
dependence between sets of random variables. We study in this paper an affinely
invariant version of the distance correlation and an empirical version of that
distance correlation, and we establish the consistency of the empirical
quantity. In the case of subvectors of a multivariate normally distributed
random vector, we provide exact expressions for the affinely invariant distance
correlation in both finite-dimensional and asymptotic settings, and in the
finite-dimensional case we find that the affinely invariant distance
correlation is a function of the canonical correlation coefficients. To
illustrate our results, we consider time series of wind vectors at the
Stateline wind energy center in Oregon and Washington, and we derive the
empirical auto and cross distance correlation functions between wind vectors at
distinct meteorological stations.
