Iterated Generalized Counting Process and its Extensions

M. Dhillon () and K. K. Kataria ()
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M. Dhillon: Indian Institute of Technology Bhilai
K. K. Kataria: Indian Institute of Technology Bhilai

Journal of Theoretical Probability, 2026, vol. 39, issue 1, 1-42

Abstract: Abstract 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.

Keywords: Bell polynomials; Martingale characterization; Iterated process; Long-range dependence property; Generalized counting process; 60J27; 60G51; 60G42; 60G55 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/s10959-025-01460-7

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