Generalization of the energy distance by Bernstein functions

We reprove the well known fact that the energy distance defines a metric on the space of Borel probability measures on a Hilbert space with finite first moment by a new approach, by analyzing the behavior of the Gaussian kernel on Hilbert spaces and a Maximum Mean Discrepancy analysis. From this new point of view we are able to generalize the energy distance metric to a family of kernels related to Bernstein functions and conditionally negative definite kernels. We also explain what occurs on the energy distance on the kernel $\|x-y\|^{\alpha}$ for every $\alpha >2$, where we also generalize the idea to a family of kernels related to derivatives of completely monotone functions and conditionally negative definite kernels.