Fractional counting process at Lévy times and its applications

Traditionally, fractional counting processes, such as the fractional Poisson process etc., have been defined using three methods: (i) through fractional differential and integral operators, (ii) by employing non-exponential waiting times in the renewal process approach, and (iii) by time-changing the Poisson process. Recently, Laskin (2024) introduced a broader class of fractional counting processes (FCP) by introducing the methodology for direct construction of the probability distribution using generalized three-parameter Mittag-Leffler function. In this paper, we introduce the time-changed fractional counting process (TCFCP), defined by time-changing the FCP with an independent Lévy subordinator. We derive distributional properties and results related to first waiting and the first passage time distribution are also discussed. We define the additive and multiplicative compound variants for the FCP and the TCFCP and examine their distributional characteristics with some typical examples. We explore some interesting connections of the TCFCP with Bell polynomials by introducing subordinated generalized fractional Bell polynomials. Finally, we present the application of the TCFCP in a shock deterioration model.