  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 correlation are analogous to product-moment
covariance and correlation, but generalize and extend these classical bivariate
measures of dependence. Distance correlation characterizes independence: it is
zero if and only if the random vectors are independent. The notion of
covariance with respect to a stochastic process is introduced, and it is shown
that population distance covariance coincides with the covariance with respect
to Brownian motion; thus, both can be called Brownian distance covariance. In
the bivariate case, Brownian covariance is the natural extension of
product-moment covariance, as we obtain Pearson product-moment covariance by
replacing the Brownian motion in the definition with identity. The
corresponding statistic has an elegantly simple computing formula. Advantages
of applying Brownian covariance and correlation vs the classical Pearson
covariance and correlation are discussed and illustrated.
