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]
If is a zero-mean multivariate normal random vector, then
More generally, if 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 .
If 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 is even, there exist (see double factorial) pair partitions of : this yields terms in the sum. For example, for order moments (i.e. random variables) there are three terms. For -order moments there are terms, and for -order moments there are terms.
We can evaluate the characteristic function of gaussians by the Isserlis theorem:
Since the formula is linear on both sides, if we can prove the real case, we get the complex case for free.
Let 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 is invertible.
Using quadratic factorization , we get
Differentiate under the integral sign with to obtain
That is, we need only find the coefficient of term in the Taylor expansion of .
If is odd, this is zero. So let , then we need only find the coefficient of term in the polynomial .
Expand the polynomial and count, we obtain the formula.
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 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. If is a vector of random variables, then
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]
If is a zero-mean multivariate normal random vector, then
More generally, if 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 .
If 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 is even, there exist (see double factorial) pair partitions of : this yields terms in the sum. For example, for order moments (i.e. random variables) there are three terms. For -order moments there are terms, and for -order moments there are terms.
We can evaluate the characteristic function of gaussians by the Isserlis theorem:
Since the formula is linear on both sides, if we can prove the real case, we get the complex case for free.
Let 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 is invertible.
Using quadratic factorization , we get
Differentiate under the integral sign with to obtain
That is, we need only find the coefficient of term in the Taylor expansion of .
If is odd, this is zero. So let , then we need only find the coefficient of term in the polynomial .
Expand the polynomial and count, we obtain the formula.
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 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. If is a vector of random variables, then