{
  "id": "1407.3072",
  "version": "v2",
  "published": "2014-07-11T08:59:24.000Z",
  "updated": "2016-07-19T08:11:24.000Z",
  "title": "Some characterizations for Markov processes as mixed renewal processes",
  "authors": [
    "N. D. Macheras",
    "S. M. Tzaninis"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from \\cite{lm6z3} (enlarging Huang's \\cite{hu} original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-11T08:59:24.000Z",
      "abstract": "For a mixed renewal process (MRP for short) with mixing parameter a random vector we prove under mild assumptions that it has the multinomial property if and only if it is a Markov process or equivalently that it is a mixed Poisson process with mixing parameter a random variable. We provide a simple example showing that our assumptions are essential for these equivalences to hold true. As a consequence the invariance of the Markov property under a certain change of measures follows. As an application we single out some concrete examples of non-trivial probability spaces, admitting MRPs and allowing us to check whether a given MRP has the Markov property or not.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-07-19T08:11:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60K05",
      "60J27",
      "28A50",
      "60A10",
      "91B30",
      "60G05"
    ],
    "keywords": [
      "mixed renewal processes",
      "markov processes",
      "characterizations",
      "markov property",
      "non-trivial probability spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3072M"
    }
  }
}