{
  "id": "2210.03981",
  "version": "v1",
  "published": "2022-10-08T09:42:15.000Z",
  "updated": "2022-10-08T09:42:15.000Z",
  "title": "Generalized Counting Process: its Non-Homogeneous and Time-Changed Versions",
  "authors": [
    "K. K. Kataria",
    "M. Khandakar",
    "P. Vellaisamy"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a non-homogeneous version of the generalized counting process (GCP), namely, the non-homogeneous generalized counting process (NGCP). We time-change the NGCP by an independent inverse stable subordinator to obtain its fractional version, and call it as the non-homogeneous generalized fractional counting process (NGFCP). A generalization of the NGFCP is obtained by time-changing the NGCP with an independent inverse subordinator. We derive the system of governing differential-integral equations for the marginal distributions of the increments of NGCP, NGFCP and its generalization. Then, we consider the GCP time-changed by a multistable subordinator and obtain its L\\'evy measure, associated Bern\\v{s}tein function and distribution of the first passage times. The GCP and its fractional version, that is, the generalized fractional counting process when time-changed by a L\\'evy subordinator are known as the time-changed generalized counting process-I (TCGCP-I) and the time-changed generalized fractional counting process-I (TCGFCP-I), respectively. We obtain the distribution of first passage times and related governing equations for the TCGCP-I. An application of the TCGCP-I to ruin theory is discussed. We obtain the conditional distribution of the $k$th order statistic from a sample whose size is modelled by a particular case of TCGFCP-I, namely, the time fractional negative binomial process. Later, we consider a fractional version of the TCGCP-I and obtain the system of differential equations that governs its state probabilities. Its mean, variance, covariance, {\\it etc.} are obtained and using which its long-range dependence property is established. Some results for its two particular cases are obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-08T09:42:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22",
      "60G55",
      "60G51",
      "91B30"
    ],
    "keywords": [
      "generalized counting process",
      "time-changed versions",
      "fractional negative binomial process",
      "fractional version",
      "first passage times"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}