{
  "id": "2009.09148",
  "version": "v1",
  "published": "2020-09-19T02:45:24.000Z",
  "updated": "2020-09-19T02:45:24.000Z",
  "title": "Characterization of Probability Distributions via Functional Equations of Power-Mixture Type",
  "authors": [
    "Chin-Yuan Hu",
    "Gwo Dong Lin",
    "Jordan M. Stoyanov"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-19T02:45:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E10",
      "60E10",
      "39B05",
      "42B10"
    ],
    "keywords": [
      "probability distribution",
      "study power-mixture type functional equations",
      "unique solutions",
      "studying distributional equations",
      "laplace-stieltjes transforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}