Generalization of the Energy Distance by Bernstein Functions

Jean Carlo Guella ()
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Jean Carlo Guella: Unicamp

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 494-521

Abstract: Abstract 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 $$\Vert x-y\Vert ^{a}$$ ‖ x - y ‖ a for every $$a >2$$ a > 2 , by describing in which circumstances it defines a distance between probabilities. We also generalize this idea to a family of kernels related to completely monotone functions of finite order and conditionally negative definite kernels.

Keywords: Energy distance; Metric spaces of strong negative type; Metrics on probabilities; Bernstein functions; Conditionally negative definite kernels; 30L05; 42A82; 43A35 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01167-z

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