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 $\|x-y\|^α$ for every $α>2$, where we also generalize the idea to a family of kernels related to derivatives of completely monotone functions and conditionally negative definite kernels.

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