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

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

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

\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, $n\in 2\mathbb {N} +1$

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

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

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

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

Generalizations

Gaussian integration by part

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

$\operatorname {E} (X_{1}f(X_{1},\ldots ,X_{n}))=\sum _{i=1}^{n}\operatorname {Cov} (X_{1}X_{i})\operatorname {E} (\partial _{X_{i}}f(X_{1},\ldots ,X_{n}))$ .

The Wick's probability formula can be recovered by induction, considering the function $f:\mathbb {R} ^{n}\to \mathbb {R}$ defined by: $f(x_{1},\ldots ,x_{n})=x_{2}\ldots x_{n}$ . 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 $(X_{1},\dots X_{n})$ is a vector of random variables, then

\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 $\{1,\ldots ,n\}$ , the product is over the blocks of $p$ and $\kappa {\big (}(X_{i})_{i\in b}{\big )}$ is the cumulants of $(X_{i})_{i\in b}$ .

References

Wick, G.C. (1950). "The evaluation of the collision matrix". Physical Review. 80 (2): 268–272. Bibcode:1950PhRv...80..268W. doi:10.1103/PhysRev.80.268.
Repetowicz, Przemysław; Richmond, Peter (2005). "Statistical inference of multivariate distribution parameters for non-Gaussian distributed time series" (PDF). Acta Physica Polonica B. 36 (9): 2785–2796. Bibcode:2005AcPPB..36.2785R.
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. arXiv:0707.3365. Bibcode:2007PhRvC..76f4314P. doi:10.1103/PhysRevC.76.064314.
Bartosch, L. (2001). "Generation of colored noise". International Journal of Modern Physics C. 12 (6): 851–855. Bibcode:2001IJMPC..12..851B. doi:10.1142/S0129183101002012.
Janson, Svante (June 1997). Gaussian Hilbert Spaces. Cambridge Core. doi:10.1017/CBO9780511526169. ISBN 9780521561280. Retrieved 2019-11-30.
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. Bibcode:2009JSP...136...89M. doi:10.1007/s10955-009-9768-3.
Isserlis, L. (1918). "On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables". Biometrika. 12 (1–2): 134–139. doi:10.1093/biomet/12.1-2.134. JSTOR 2331932.
Isserlis, L. (1916). "On Certain Probable Errors and Correlation Coefficients of Multiple Frequency Distributions with Skew Regression". Biometrika. 11 (3): 185–190. doi:10.1093/biomet/11.3.185. JSTOR 2331846.
Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent (2019-11-01). "Local Conformal Structure of Liouville Quantum Gravity". Communications in Mathematical Physics. 371 (3): 1005–1069. arXiv:1512.01802. Bibcode:2019CMaPh.371.1005K. doi:10.1007/s00220-018-3260-3. ISSN 1432-0916.
Remy, Guillaume (2017-10-18). "The Fyodorov-Bouchaud formula and Liouville conformal field theory". arXiv:1710.06897 [math.PR].
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.

Isserlis' theorem