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Isserlis' Theorem

Isserlis' theorem expresses every joint moment of a centered multivariate normal distribution as a sum of products of covariances (Isserlis 1918, Janson 1997). Let X_1, X_2, ..., X_n have zero mean, and write C_(ij)=E[X_iX_j]. If P_2(2m) is the set of set partitions of {1,2,...,2m} into pairs, then

E[product_(i=1)^nX_i]={0 if n is odd; sum_(pi in P_2(2m))product_({i,j} in pi)C_(ij) if n=2m .
(1)

The number of these set partitions is (2m-1)!!, where z!! denotes the double factorial of z. For example,

E[X_1X_2X_3X_4]=C_(12)C_(34)+C_(13)C_(24)+C_(14)C_(23).
(2)

For a multivariate normal distribution with nonzero means, the theorem applies after replacing each X_i by X_i-E[X_i] and expanding. The Wick-Isserlis formula is the corresponding pairing formulation of the theorem.

See also

Covariance Matrix, Expectation Value, Gaussian Distribution, Moment, Multivariate Normal Distribution, Set Partition, Wick-Isserlis Formula

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References

Isserlis, L. "On a Formula for the Product-Moment Coefficient of Any Order of a Normal Frequency Distribution in Any Number of Variables." Biometrika 12, 134-139, 1918. https://doi.org/10.1093/biomet/12.1-2.134.Janson, S. Gaussian Hilbert Spaces. Cambridge, England: Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511526169.

Cite this as:

Weisstein, Eric W. "Isserlis' Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IsserlisTheorem.html

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