= Isserlis theorem
{c}
{wiki=Isserlis'_theorem}
= Gaussian moment pairing theorem
{c}
{synonym}
For centered jointly <Gaussian random variables>, an <odd> product has zero expectation, and an <even> product's expectation is the sum over all pairings of the products of the pair covariances. In particular,
$$
\mathbb E[X_1X_2X_3X_4]
=\mathbb E[X_1X_2]\mathbb E[X_3X_4]
+\mathbb E[X_1X_3]\mathbb E[X_2X_4]
+\mathbb E[X_1X_4]\mathbb E[X_2X_3].
$$
For a proof, the <moment-generating function> is $\exp(t^T\Sigma t/2)$. Differentiating once in each variable at zero leaves precisely products of covariance entries indexed by complete pairings. No <independence> assumption between the coordinates is needed.
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