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In this post there is a clever proof of Isserlis' theorem, but it is based on a identity for the Laplace transform that I do not know how to prove. Why is it that, given a random vector $X,$ it holds that $E(e^{s^\prime X})=e^{\frac{1}{2}s^\prime C s}$? (Here, as in the above-mentioned post, $s^\prime$ is the transpose vector of $s$ and $C$ is the covariance Matrix of $X$)

Qwertuy
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    The post referred to is about jointly Gaussian random variables. If you know that characteristic function of jointly Gaussian random variables you can get the Laplace transform using complex analysis. – Kavi Rama Murthy May 05 '18 at 12:08
  • yeap, you are right... for multivariate normal distributions the identity follows from the typical gaussian integral. It was a silly question, thanks :S – Qwertuy May 05 '18 at 12:48

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