Copula Relations in Compound Poisson Processes

TL;DR

This paper studies how the dependence structure of jump distributions in multidimensional compound Poisson processes influences the overall process, showing convergence to a Gaussian copula under certain conditions and validating findings with simulations.

Contribution

It demonstrates the convergence of copulas in compound Poisson processes to a Gaussian copula and explores the relationship between jump distribution dependence and process dependence.

Findings

Copula of CPP converges to a Gaussian copula asymptotically.

The equation relating the copula of the process to the jump distribution generally does not hold.

Simulation results support the theoretical convergence and dependence structure findings.

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.