Isserlis' theorem


In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.
This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of. Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise.

Statement

If is a zero-mean multivariate normal random vector, thenwhere the sum is over all the pairings of, i.e. all distinct ways of partitioning into pairs, and the product is over the pairs contained in.
In his original paper, Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the order moments, which takes the appearance
Odd case, n\in 2\mathbb{N}+1
If is odd, there does not exist any pairing of. Under this hypothesis, Isserlis' theorem implies that:
Even case, n\in 2\mathbb{N}
If is even, there exist pair partitions of : this yields terms in the sum. For example, for order moments there are three terms. For -order moments there are terms, and for -order moments there are terms.

Generalizations

Gaussian integration by part

An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts. If is a zero-mean multivariate normal random vector, then
The Wick's probability formula can be recovered by induction, considering the function defined by:. Among other things, this formulation is important in Liouville Conformal Field Theory to obtain conformal Ward's identities, BPZ equations and to prove the Fyodorov-Bouchaud formula.

Non-Gaussian random variables

For non-Gaussian random variables, the moment-cumulants formula replaces the Wick's probability formula. If is a vector of random variables, then where the sum is over all the partitions of, the product is over the blocks of and is the cumulants of.