Multivariate Generalized Counting Process via Gamma Subordination

TL;DR

The paper develops a multivariate gamma subordinator built from independent gamma processes time-changed by an independent negative binomial clock and uses it to time-change a multivariate generalized counting process (GCP). It provides explicit formulas for the subordinator's joint Laplace-Stieltjes transform, pdf, and governing differential equations, and characterizes its covariance and codifference. The authors then study the time-changed multivariate GCP, obtaining the joint distributional objects (state probabilities, pgf, and Lévy measure) and a differential equation for the pgf, along with a practical hazard-rate framework. Finally, they apply the framework to a shock-model scenario, deriving the survival function of the system failure time and offering closed forms for Geometric and Hypergeometric threshold cases, highlighting potential applications in reliability and risk analysis.

Abstract

In this paper, we study a multivariate gamma subordinator whose components are independent gamma processes subject to a random time governed by an independent negative binomial process. We derive the explicit expressions for its joint Laplace-Stieltjes transform, its probability density function and the associated governing differential equations. Also, we study a time-changed variant of the multivariate generalized counting process where the time is changed by an independent multivariate gamma subordinator. For this time-changed process, we obtain the corresponding Lévy measure and probability mass function. Later, we discuss an application of the time-changed multivariate generalized counting process to a shock model.

Figures (4)

• Figure 1: Plots 1(a), 1(b) represent the survival function \ref{['Geo']} and \ref{['hypergeo']} for the parameter values $\theta=0.5$, $a_1=a_2=1$, $\lambda_1=0.5$, $\lambda_{21}=0.5$, $\lambda_{22}=0.5$ and: $p=0.5$ for 1(a) and $N=2$, $K=1$, $n=1$ for 1(b) respectively.
• Figure 2: Plots 2(a), 2(b) represent the survival function \ref{['Geo']} and \ref{['hypergeo']} for the parameter values $\theta=0.5$, $a_1=a_2=1$, $\lambda_1=1$, $\lambda_{21}=1$, $\lambda_{22}=1$ and: $p=0.5$ for 2(a) and $N=2$, $K=1$, $n=1$ for 2(b) respectively.
• Figure 3: Plots 3(a), 3(b) represents the survival function \ref{['Geo']} and \ref{['hypergeo']} for the parameter values $\theta=0.5$, $a_1=a_2=1$, $\lambda=1$, $\lambda_1=0.5$, $\lambda_{21}=0.5$, $\lambda_{22}=0.5$ and $K=1$, $n=1$ (for 3(b)) respectively.
• Figure 4: Plots 4(a), 4(b) represents the survival function \ref{['Geo']} and \ref{['hypergeo']} for the parameter values $\theta=0.5$, $a_1=a_2=1$, $\lambda=1$, $\lambda_1=1$, $\lambda_{21}=1$, $\lambda_{22}=1$ and $K=1$, $n=1$ (for 4(b)) respectively.

Theorems & Definitions (17)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• proof
• proof