On the uniqueness of distance covariance

Distance covariance and distance correlation are non-negative real numbers that characterize the independence of random vectors in arbitrary dimensions. In this work we prove that distance covariance is unique, starting from a definition of a covariance as a weighted L2 norm that measures the distance between the joint characteristic function of two random vectors and the product of their marginal characteristic functions. Rigid motion invariance and scale equivariance of these weighted L2 norms imply that the weight function of distance covariance is unique.