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.
is a zero-mean multivariate normal random vector, then
, i.e. all distinct ways of partitioning
, and the product is over the pairs contained in
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 of the covariance matrix of
is odd, there does not exist any pairing of
Under this hypothesis, Isserlis' theorem implies that
In his original paper,[7] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the
order moments,[8] which takes the appearance If
(see double factorial) pair partitions of
random variables) there are three terms.
We can evaluate the characteristic function of gaussians by the Isserlis theorem:
be the covariance matrix, so that we have the zero-mean multivariate normal random vector
Since both sides of the formula are continuous with respect to
, it suffices to prove the case when
Differentiate under the integral sign with
That is, we need only find the coefficient of term
, then we need only find the coefficient of term
Expand the polynomial and count, we obtain the formula.
An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts.
is a zero-mean multivariate normal random vector, then
The Wick's probability formula can be recovered by induction, considering the function
Among other things, this formulation is important in Liouville conformal field theory to obtain conformal Ward identities, BPZ equations[9] and to prove the Fyodorov-Bouchaud formula.
[10] For non-Gaussian random variables, the moment-cumulants formula[11] replaces the Wick's probability formula.
uniformly distributed on the unit sphere
These results are discussed in the context of random vectors and irreducible representations in the work by Kushkuley (2021).