(while in Eq.(\[MCEInt\]) they are), but superpositions of oscillating and decaying parts. Thus we extent the limits of integration to $\pm \infty$, what introduces a negligible error. Eq.(\[MCEInt\]) therefore simplifies to: $$\begin{aligned}
\label{MCEIntApprox}
\mathcal{Z}^{Q^j} &\simeq&   \left[ \prod_{j=1}^{6} \int
  \limits_{-\infty}^{\infty} \frac{d\phi_j}{\left( 2\pi \right)} \right] ~\exp
\Big[  -iQ^j\phi_j
~+~V \sum_{n=0}^{\infty} \frac{i^n}{n!} \; \kappa_n^{j_1,j_2,\dots,j_n } \;
\phi_{j_1} \phi_{j_2} \dots \phi_{j_n} \Big] ~.\end{aligned}$$ Summation over repeated indices is implied. Existence and finiteness of the first three cumulants provided, any such integral can be shown to converge to a multivariate normal distribution in the large volume limit: $$\label{MCE_MultNormal}
\mathcal{Z}^{Q^j} ~\simeq~ Z_{gce} \frac{\exp \left(-\frac{\xi^j \; \xi_{ j}}{2}
  \right)}{\left(2\pi V \right)^{6/2} \det| \sigma| }~,$$ where $Z_{gce}\equiv \exp \left[V \kappa_0 \right]$ is the GCE partition function, $\kappa_0$ is the cumulant of $0^{th}$ order, $\xi^j=\left( Q^k - V
  \kappa_1^k \right) \left( \sigma^{-1}\right)_{k}^{\;\;j}  V^{-1/2}~$ is a measure for the distance of a particular macro state $Q^k$ to the peak $V
\kappa_1^k$ of the joint distribution, and $\sigma$ is the square root of the second rank tensor $\kappa_2$, see [@clt] for details.

The normalization in Eq.(\[MCEprob2\]) can essentially be found in two ways. The first way would be to integrate the distribution (\[MCE\_MultNormal\]) over all possible values of multiplicity $N_{\Omega}$, while 