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**akbar** Let $\displaystyle (\xi_1,...,\xi_{2n})$ a Gaussian vector. Assuming that $\displaystyle \mathbb{E}\xi_i=0, 1\leq i \leq 2n$, how to do you prove that:

$\displaystyle \mathbb{E}(\xi_1\xi_2...\xi_{2n})=\Sigma_\sigma\ma thbb{E}(\xi_{\sigma_1}\xi_{\sigma_2})...\mathbb{E} (\xi_{\sigma_{2n-1}}\xi_{\sigma_{2n}})$

where $\displaystyle \sigma=((\sigma_1,\sigma_2),...,(\sigma_{2n-1},\sigma_{2n})), 1\leq \sigma_i \leq 2n$ is a partition of {1,...,2n} into n pairs, and the summation extends over all the partitions (the permutation of elements of a pari is considered to yied to the same partition)?