  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.
