Isserlis's theorem on arvau

Wikipedia / Isserlis's theorem

In probability theory, Isserlis's 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

is a zero-mean multivariate normal.

is a zero-mean complex-valued multivariate normal random vector, then the formula still holds.

The expression on the right-hand side is also known as the hafnian.

Odd case

Even case

In his original paper, Leon Isserlis proves this theorem by mathematical induction order moments, which takes the appearance

(see double factorial terms.

Example

We can evaluate the characteristic function

Proof

, if we can prove the real case, we get the complex case for free.

be the covariance matrix, so that we have the zero-mean multivariate normal.

, we get

Differentiate under the integral sign.

Generalizations

Gaussian integration by parts

An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts is a zero-mean multivariate normal random vector, then

This is a generalization of Stein's lemma.

. Among other things, this formulation is important in Liouville conformal field theory to obtain conformal Ward identities, BPZ equations and to prove the Fyodorov-Bouchaud formula.

Non-Gaussian random variables

For non-Gaussian random variables, the moment-cumulants formula is a vector of random variableswhere the sum is over all the partitions is the joint cumulant.

Uniform distribution on the unit sphere

almost surely. In this setting, the following holds.

is the Kronecker delta.

denotes the double factorial.

These results are discussed in the context of random vectors and irreducible representations in the work by Kushkuley (2021).