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<p>In this paper, we study the composition of two independent generalized counting processes (GCPs) which we call the iterated generalized counting process (IGCP). Its distributional properties such as the transition probabilities, probability generating function, state probabilities, and corresponding Lévy measure are obtained. We study some of its extensions, for example, the compound IGCP, the multivariate IGCP, and the <i>q</i>-iterated GCP. Also, we study some integrals of the IGCP. It is shown that the IGCP and the compound IGCP are identically distributed to a compound GCP, which lead to their martingale characterizations. Later, a time-changed variant of the IGCP is considered where the time is changed by an inverse stable subordinator. Using its covariance structure, we establish that the time-changed IGCP exhibits a long-range dependence property. Moreover, we show that its increment process exhibits a short-range dependence property. Also, it is shown that its one-dimensional distributions are not infinitely divisible. Additionally, some of its potential real-life applications are discussed.</p>

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Iterated Generalized Counting Process and its Extensions

  • 期刊论文

  • 发表日期：

    2025年11月15日

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  • M. Dhillon,
  • K. K. Kataria

摘要

In this paper, we study the composition of two independent generalized counting processes (GCPs) which we call the iterated generalized counting process (IGCP). Its distributional properties such as the transition probabilities, probability generating function, state probabilities, and corresponding Lévy measure are obtained. We study some of its extensions, for example, the compound IGCP, the multivariate IGCP, and the q-iterated GCP. Also, we study some integrals of the IGCP. It is shown that the IGCP and the compound IGCP are identically distributed to a compound GCP, which lead to their martingale characterizations. Later, a time-changed variant of the IGCP is considered where the time is changed by an inverse stable subordinator. Using its covariance structure, we establish that the time-changed IGCP exhibits a long-range dependence property. Moreover, we show that its increment process exhibits a short-range dependence property. Also, it is shown that its one-dimensional distributions are not infinitely divisible. Additionally, some of its potential real-life applications are discussed.