# Generalized Fractional Counting Process

Source: [Generalized Fractional Counting Process](<https://www.sota2.com/research/paper/68fa5a0e1e209131ce9bc82a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T16:16:42.236Z

Authors: K. K. Kataria, M. Khandakar

Published: 2021-06-22T14:44:51.000Z

## Abstract

In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. (2016). For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied using which its long-range dependence property is established. It is shown that the increments of GFCP exhibits the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP) is discussed for which we obtain a limiting result, a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order $k$, the P\\'olya-Aeppli process of order $k$, the negative binomial process and their fractional versions etc. are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2106.11833>)

[PDF](<https://arxiv.org/pdf/2106.11833.pdf>)