# Isserlis' theorem

In probability theory, Isserlis’ theorem or Wick’s 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 particularly important in particle physics, where it is known as Wick's theorem after the work of Wick (1950). Other applications include the analysis of portfolio returns,[1] quantum field theory[2] and generation of colored noise.[3]

## Theorem statement

### The Isserlis theorem

If ${\displaystyle (X_{1},\dots X_{2n})}$ is a zero-mean multivariate normal random vector, then

{\displaystyle {\begin{aligned}&\operatorname {E} [\,X_{1}X_{2}\cdots X_{2n}\,]=\sum \prod \operatorname {E} [\,X_{i}X_{j}\,]=\sum \prod \operatorname {Cov} (\,X_{i},X_{j}\,),\\&\operatorname {E} [\,X_{1}X_{2}\cdots X_{2n-1}\,]=0,\end{aligned}}}

where the notation ∑ ∏ means summing over all distinct ways of partitioning X1, …, X2n into pairs Xi,Xj and each summand is the product of the n pairs.[4] This yields ${\displaystyle (2n)!/(2^{n}n!)=(2n-1)!!}$ terms in the sum (see double factorial). For example, for fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms (as you can check in the examples below).

In his original paper,[5] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the fourth-order moments,[6] which takes the appearance

${\displaystyle \operatorname {E} [\,X_{1}X_{2}X_{3}X_{4}\,]=\operatorname {E} [X_{1}X_{2}]\,\operatorname {E} [X_{3}X_{4}]+\operatorname {E} [X_{1}X_{3}]\,\operatorname {E} [X_{2}X_{4}]+\operatorname {E} [X_{1}X_{4}]\,\operatorname {E} [X_{2}X_{3}].}$

For sixth-order moments, Isserlis' theorem is:

{\displaystyle {\begin{aligned}&\operatorname {E} [X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\\[5pt]={}&\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{4}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{5}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{6}]\operatorname {E} [X_{4}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{4}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{3}X_{6}]+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{3}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{3}X_{6}]+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{3}X_{4}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{4}X_{5}]+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{3}X_{5}]+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{3}X_{4}].\end{aligned}}}