On distance covariance in metric and Hilbert spaces

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