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,For a proof, the moment-generating function is . 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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Isserlis' theorem, also known as the Isserlis-Wick theorem, is a fundamental result in probability theory and statistics, particularly in the context of Gaussian random variables. It provides a way to compute the expected value of products of even numbers of Gaussian random variables.