On distance covariance in metric and Hilbert spaces

Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see Sz\'ekely, Rizzo and Bakirov (2007) and Lyons (2013). It is known that the distance covariance, and its generalization $\alpha$-distance covariance, can be defined in several different ways that are equivalent under some moment conditions. The present paper considers four such definitions and find minimal moment conditions for each of them, together with some partial results when these conditions are not satisfied. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their ($\alpha$-)distance covariance is 0; this extends results by Lyons (2013) and Dehling et al. (2018+). The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Sz\'ekely, Rizzo and Bakirov (2007).