Copula Relations in Compound Poisson Processes

Abstract

We investigate in multidimensional compound Poisson processes (CPP) the relation between the dependence structure of the jump distribution and the dependence structure of the respective components of the CPP itself. For this purpose the asymptotic $\lambda t\to \infty$ is considered, where $\lambda$ denotes the intensity and $t$ the time point of the CPP. For modeling the dependence structures we are using the concept of copulas. We prove that the copula of a CPP converges under quite general assumptions to a specific Gaussian copula, depending on the underlying jump distribution. Let $F$ be a $d$-dimensional jump distribution $(d\geq 2)$, $\lambda>0$ and let $\Psi(\lambda,F)$ be the distribution of the corresponding CPP with intensity $\lambda$ at the time point $1$. Further, denote the operator which maps a $d$-dimensional distribution on its copula as $\mathcal{T}$. The starting point of our investigation was the validity of the equation \begin{equation} \label{marFreeEq} \mathcal{T}(\Psi(\lambda,F))=\mathcal{T}(\Psi(\lambda,\mathcal{T}F)). \end{equation} Our asymptotic theory implies that this equation is, in general, not true. A simulation study that confirms our theoretical results is given in the last section.