# Some Time-changed fractional Poisson processes

## TL;DR

This paper investigates two types of fractional Poisson processes time-changed by Lévy subordinators, analyzing their distributional properties, dependence structures, and providing simulations to illustrate their behaviors.

## Contribution

It introduces and studies TCFPP-I and TCFPP-II, new fractional Poisson processes with time changes, detailing their properties, distributions, and differential equations, which were not previously explored.

## Key findings

- TCFPP-I exhibits long-range dependence under certain conditions.
- TCFPP-II is characterized as a renewal process with a specific waiting time distribution.
- Sample path simulations demonstrate the processes' behaviors.

## Abstract

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent L\'evy subordinator and the inverse of the L\'evy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. Its bivariate distributions and also the governing difference-differential equation are derived. Some specific examples for both the processes are discussed. Finally, we present the simulations of the sample paths of these processes.