# Thread: Gaussian vector and expectation

1. ## Gaussian vector and expectation

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)?

The natural approach is to prove it by induction using the usual results on (partial) derivatives of the characteristic function and the moments, assuming the Gaussian vector is represented using a correlation matrix. However, the calculation, albeit tedious, cannot be concluded without some results on permutation to carry on the induction hypothesis.

This has probably to do with decomposition of permutations into cycles, so I was wondering if anyone had any idea about how to proceed further.

2. Originally Posted by 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)?
This is called Wick's formula. You can find a proof here on p.111, for instance (this course is a very interesting one, by the way...). This proof may not be the simplest (but I knew it was there, that's why I refered to it); using the keyword "wick formula", you should be able to find plenty of other references.

3. Thanks for the keyword (been looking for that one).

The proof in the course is actually not that hard. It is essentially based on the result on linear combinations of Gaussian variables and the permutations obtained through powers of multinomials (like for the definition of the signature of a permutation). Doing it by induction was finally not leading anywhere...

The course looks very interesting indeed. Thanks for the link.