{
  "id": "2102.00633",
  "version": "v1",
  "published": "2021-02-01T04:44:41.000Z",
  "updated": "2021-02-01T04:44:41.000Z",
  "title": "Generalization of the energy distance by Bernstein functions",
  "authors": [
    "Jean Carlo Guella"
  ],
  "categories": [
    "math.FA",
    "math.CA",
    "math.PR"
  ],
  "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 $\\|x-y\\|^{\\alpha}$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T04:44:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A82",
      "43A35"
    ],
    "keywords": [
      "bernstein functions",
      "conditionally negative definite kernels",
      "hilbert space",
      "generalization",
      "maximum mean discrepancy analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}