On the Multivariate Generalized Counting Process and its Time-Changed Variants

TL;DR

This paper introduces a multivariate generalized counting process and explores its time-changed variants using different subordinators, deriving their distributional properties, Le9vy characteristics, and applications to shock models.

Contribution

It presents a novel multivariate generalized counting process and analyzes its time-changed versions with explicit distributional and dependence properties, including applications.

Findings

Derived probability generating functions and differential equations for the variants.

Identified Le9vy processes among the time-changed variants.

Provided explicit covariance and codifference expressions for component processes.

Abstract

In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator $etc.$ Several distributional properties that include the probability generating function, probability mass function and their governing differential equations are obtained for these variants. It is shown that some of these time-changed processes are L\'evy and for such processes we have derived the associated L\'evy measure. The explicit expressions for the covariance and codifference of the component processes for some of these time-changed variants are obtained. An application of the multivariate generalized space fractional counting process to shock models is discussed.