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 Wick (1950).^[1] Other applications include the analysis of portfolio returns,^[2] quantum field theory^[3] and generation of colored noise.^[4]

Statement[]

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

where the sum is over all the pairings of ${\displaystyle \{1,\ldots ,n\}}$, i.e. all distinct ways of partitioning ${\displaystyle \{1,\ldots ,n\}}$ into pairs ${\displaystyle \{i,j\}}$, and the product is over the pairs contained in ${\displaystyle p}$.^[5][6]

In his original paper,^[7] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the ${\displaystyle 4^{\text{th}}}$ order moments,^[8] which takes the appearance

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

Odd case, ${\displaystyle n\in 2\mathbb {N} +1}$[]

If ${\displaystyle n=2m+1}$ is odd, there does not exist any pairing of ${\displaystyle \{1,\ldots ,2m+1\}}$. Under this hypothesis, Isserlis' theorem implies that

This also follows from the fact that ${\displaystyle -X=(-X_{1},\dots ,-X_{n})}$ has the same distribution as ${\displaystyle X}$, which implies that ${\displaystyle \operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=\operatorname {E} [\,(-X_{1})\cdots (-X_{2m+1})\,]=-\operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=0}$.

Even case, ${\displaystyle n\in 2\mathbb {N} }$[]

If ${\displaystyle n=2m}$ is even, there exist ${\displaystyle (2m)!/(2^{m}m!)=(2m-1)!!}$ (see double factorial) pair partitions of ${\displaystyle \{1,\ldots ,2m\}}$: this yields ${\displaystyle (2m)!/(2^{m}m!)=(2m-1)!!}$ terms in the sum. For example, for ${\displaystyle 4^{\text{th}}}$ order moments (i.e. ${\displaystyle 4}$ random variables) there are three terms. For ${\displaystyle 6^{\text{th}}}$-order moments there are ${\displaystyle 3\times 5=15}$ terms, and for ${\displaystyle 8^{\text{th}}}$-order moments there are ${\displaystyle 3\times 5\times 7=105}$ terms.

Generalizations[]

Gaussian integration by parts[]

An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts. If ${\displaystyle (X_{1},\dots X_{n})}$ is a zero-mean multivariate normal random vector, then

The Wick's probability formula can be recovered by induction, considering the function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ defined by ${\displaystyle f(x_{1},\ldots ,x_{n})=x_{2}\ldots x_{n}}$. Among other things, this formulation is important in Liouville Conformal Field Theory to obtain , ^[9] and to prove the .^[10]

Non-Gaussian random variables[]

For non-Gaussian random variables, the moment-cumulants formula^[11] replaces the Wick's probability formula. If ${\displaystyle (X_{1},\dots X_{n})}$ is a vector of random variables, then

where the sum is over all the partitions of ${\displaystyle \{1,\ldots ,n\}}$, the product is over the blocks of ${\displaystyle p}$ and ${\displaystyle \kappa {\big (}(X_{i})_{i\in b}{\big )}}$ is the cumulant of ${\displaystyle (X_{i})_{i\in b}}$.

See also[]

• Wick's theorem
• Cumulants
• Normal distribution

References[]

Further reading[]

• Koopmans, Lambert G. (1974). The spectral analysis of time series. San Diego, CA: Academic Press.