IsserlisIsserls

IsserlisIsserls is a term sometimes used as a variant spelling of Isserlis' theorem, a fundamental result in probability theory about Gaussian moments. The theorem provides a concise way to compute the expectation of a product of a finite set of jointly Gaussian random variables in terms of their second moments.

Formally, let X1, X2, ..., X2n be jointly Gaussian with means μi and covariance Cov(Xi, Xj) = E[(Xi − μi)(Xj − μj)]. Then the mixed moment E[X1 X2 ... X2n] equals the sum over all pairings of {1, ..., 2n} of the products of covariances for each pair. In the common zero-mean case, where μi = 0 for all i, this reduces to E[X1 X2 ... X2n] = sum over pairings ∏ Cov(Xa, Xb). If the total number of factors is odd, the expectation is zero. The number of pairings is (2n − 1)!!.

The theorem is a Gaussian moment factoring rule and is closely related to Wick's theorem in physics, where it is used to evaluate expectations of products of Gaussian fields. It has wide applications in statistics, random matrix theory, time series analysis, and probabilistic proofs.

The specific spelling IsserlisIsserls may reflect typographical variants or informal references. In standard mathematical literature, the result is commonly called Isserlis' theorem or Isserlis' formula, with some texts also noting its connection to Wick's theorem in physics. The essential content remains the same: higher-order moments of Gaussian variables can be expressed entirely through second-order moments.