Generalized Fractional Counting Process

Kataria, K. K.; Khandakar, M.

doi:10.1007/s10959-022-01160-6

Cited by 10 publications

(19 citation statements)

References 21 publications

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“…. , k, the GCP reduces to the Poisson process of order k (see [19,Section 4.1]). Thus, for such λ j , the MFCP reduces to a fractional version of the Poisson process of order k. Also, for…”

Section: Mixed Fractional Counting Processmentioning

confidence: 99%

“…For , the MFCP reduces to the MFPP (see [1], [3]) as in this case the GCP reduces to the Poisson process. On taking for all , the GCP reduces to the Poisson process of order k (see [19, Section 4 1…”

Section: Mixed Fractional Counting Processmentioning

confidence: 99%

“…Thus, for such , the MFCP reduces to a fractional version of the Poisson process of order k . Also, for , , , the GCP reduces to the Pólya–Aeppli process of order k (see [19, Section 4 2…”

Section: Mixed Fractional Counting Processmentioning

confidence: 99%

“…On substituting and in Proposition 3.3, we get the r th factorial moment of the GFCP (see [19, eq. (19)]).…”

Section: Mixed Fractional Counting Processmentioning

confidence: 99%

“…For more properties of the GFCP and an application of the GCP in risk theory, we refer the reader to Kataria and Khandakar [19]. Also, for other counting processes that perform jumps of amplitude larger than 1, we refer to Orsingher and Polito [24] and Orsingher and Toaldo [25].…”

Section: Introductionmentioning

confidence: 99%

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On a time-changed variant of the generalized counting process

Khandakar,

Kataria

2023

J. Appl. Probab.

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In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

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Section: Mixed Fractional Counting Processmentioning

confidence: 99%