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$\displaystyle{ \operatorname{E} [\,X_1 X_2\cdots X_{n}\,] = \sum_{p\in P_n^2}\prod_{\{i,j\}\in p} \operatorname{E}[\,X_i X_j\,] = \sum_{p\in P_n^2}\prod_{\{i,j\}\in p} \operatorname{Cov}(\,X_i, X_j\,), }$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

$\displaystyle{ \operatorname{E}[\,X_1 X_2 X_3 X_4\,] = \operatorname{E}[X_1X_2]\,\operatorname{E}[X_3X_4] + \operatorname{E}[X_1X_3]\,\operatorname{E}[X_2X_4] + \operatorname{E}[X_1X_4]\,\operatorname{E}[X_2X_3]. }$

Odd case

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$\displaystyle{ \operatorname{E}[\,X_1 X_2\cdots X_{2m+1}\,] = 0. }$ 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

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 $\displaystyle{ 3\times5\times7 = 105 }$ terms.

Proof

Let $\displaystyle{ \Sigma_{ij} = \operatorname{Cov}(X_i, X_j) }$ be the covariance matrix, so that we have the zero-mean multivariate normal random vector $\displaystyle{ (X_1, ..., X_n) \sim N(0, \Sigma) }$ .

Using quadratic factorization $\displaystyle{ -x^T\Sigma^{-1}x/2 + v^Tx - v^T\Sigma v/2 = -(x-\Sigma v)^T\Sigma^{-1}(x-\Sigma v)/2 }$, we get

$\displaystyle{ \frac{1}{\sqrt{(2\pi)^n\det\Sigma}}\int e^{-x^T\Sigma^{-1}x/2 + v^Tx} dx = e^{v^T\Sigma v/2} }$

Differentiate under the integral sign with $\displaystyle{ \partial_{v_1, ..., v_n}|_{v_1, ..., v_n=0} }$ to obtain

$\displaystyle{ E[X_1\cdots X_n] = \partial_{v_1, ..., v_n}|_{v_1, ..., v_n=0}e^{v^T\Sigma v/2} }$.

That is, we need only find the coefficient of term $\displaystyle{ v_1\cdots v_n }$ in the Taylor expansion of $\displaystyle{ e^{v^T\Sigma v/2} }$.

If $\displaystyle{ n }$ is odd, this is zero. So let $\displaystyle{ n = 2m }$, then we need only find the coefficient of term $\displaystyle{ v_1\cdots v_n }$ in the polynomial $\displaystyle{ \frac{1}{m!}(v^T\Sigma v/2)^m }$.

Expand the polynomial and count, we obtain the formula. $\displaystyle{ \square }$

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

$\displaystyle{ \operatorname{E}(X_1 f(X_1,\ldots,X_n))=\sum_{i=1}^{n} \operatorname{Cov}(X_1X_i)\operatorname{E}(\partial_{X_i}f(X_1,\ldots,X_n)). }$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 conformal Ward's identities, BPZ equations[9] and to prove the Fyodorov-Bouchaud formula.[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 $\displaystyle{ \operatorname{E}(X_1 \ldots X_n)=\sum_{p\in P_n} \prod_{b\in p} \kappa\big((X_i)_{i\in b}\big), }$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 joint cumulant of $\displaystyle{ (X_i)_{i\in b} }$.

References

1. Wick, G.C. (1950). "The evaluation of the collision matrix". Physical Review 80 (2): 268–272. doi:10.1103/PhysRev.80.268. Bibcode1950PhRv...80..268W.
2. Repetowicz, Przemysław; Richmond, Peter (2005). "Statistical inference of multivariate distribution parameters for non-Gaussian distributed time series". Acta Physica Polonica B 36 (9): 2785–2796. Bibcode2005AcPPB..36.2785R.
3. Perez-Martin, S.; Robledo, L.M. (2007). "Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem". Physical Review C 76 (6): 064314. doi:10.1103/PhysRevC.76.064314. Bibcode2007PhRvC..76f4314P.
4. Bartosch, L. (2001). "Generation of colored noise". International Journal of Modern Physics C 12 (6): 851–855. doi:10.1142/S0129183101002012. Bibcode2001IJMPC..12..851B.
5. Janson, Svante (June 1997). Gaussian Hilbert Spaces. doi:10.1017/CBO9780511526169. ISBN 9780521561280. Retrieved 2019-11-30.
6. Michalowicz, J.V.; Nichols, J.M.; Bucholtz, F.; Olson, C.C. (2009). "An Isserlis' theorem for mixed Gaussian variables: application to the auto-bispectral density". Journal of Statistical Physics 136 (1): 89–102. doi:10.1007/s10955-009-9768-3. Bibcode2009JSP...136...89M.
7. Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent (2019-11-01). "Local Conformal Structure of Liouville Quantum Gravity". Communications in Mathematical Physics 371 (3): 1005–1069. doi:10.1007/s00220-018-3260-3. ISSN 1432-0916. Bibcode2019CMaPh.371.1005K.
8. Remy, Guillaume (2020). "The Fyodorov–Bouchaud formula and Liouville conformal field theory". Duke Mathematical Journal 169. doi:10.1215/00127094-2019-0045.
9. Leonov, V. P.; Shiryaev, A. N. (January 1959). "On a Method of Calculation of Semi-Invariants". Theory of Probability & Its Applications 4 (3): 319–329. doi:10.1137/1104031.