Isserlis' theorem

In probability theory, Isserlis’ theorem or Wick’s 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 particularly important in particle physics, where it is known as Wick's theorem after the work of Wick (1950). Other applications include the analysis of portfolio returns,^[1] quantum field theory^[2] and generation of colored noise.^[3]

Theorem statement

The Isserlis theorem

If $(X_{1},\dots X_{2n})$ is a zero-mean multivariate normal random vector, then

{\begin{aligned}&\operatorname {E} [\,X_{1}X_{2}\cdots X_{2n}\,]=\sum \prod \operatorname {E} [\,X_{i}X_{j}\,]=\sum \prod \operatorname {Cov} (\,X_{i},X_{j}\,),\\&\operatorname {E} [\,X_{1}X_{2}\cdots X_{2n-1}\,]=0,\end{aligned}}

where the notation ∑ ∏ means summing over all distinct ways of partitioning X₁, …, X_2n into pairs X_i,X_j and each summand is the product of the n pairs.^[4] This yields $(2n)!/(2^{n}n!)=(2n-1)!!$ terms in the sum (see double factorial). For example, for fourth order moments (four variables) there are three terms. For sixth-order moments there are 3 × 5 = 15 terms, and for eighth-order moments there are 3 × 5 × 7 = 105 terms (as you can check in the examples below).

In his original paper,^[5] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the fourth-order moments,^[6] 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}].

For sixth-order moments, Isserlis' theorem is:

{\begin{aligned}&\operatorname {E} [X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]\\[5pt]={}&\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{4}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{5}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{2}]\operatorname {E} [X_{3}X_{6}]\operatorname {E} [X_{4}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{3}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{4}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{5}X_{6}]+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{3}X_{6}]+\operatorname {E} [X_{1}X_{4}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{3}X_{5}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{4}X_{6}]+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{3}X_{6}]+\operatorname {E} [X_{1}X_{5}]\operatorname {E} [X_{2}X_{6}]\operatorname {E} [X_{3}X_{4}]\\[5pt]&{}+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{3}]\operatorname {E} [X_{4}X_{5}]+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{4}]\operatorname {E} [X_{3}X_{5}]+\operatorname {E} [X_{1}X_{6}]\operatorname {E} [X_{2}X_{5}]\operatorname {E} [X_{3}X_{4}].\end{aligned}}

References

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.
Isserlis, L. (1916). "On Certain Probable Errors and Correlation Coefficients of Multiple Frequency Distributions with Skew Regression". Biometrika. 11: 185–190. doi:10.1093/biomet/11.3.185. JSTOR 2331846.
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: 134–139. doi:10.1093/biomet/12.1-2.134. JSTOR 2331932.
Koopmans, Lambert G. (1974). The spectral analysis of time series. San Diego, CA: Academic Press.
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.
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. arXiv:0707.3365. Bibcode:2007PhRvC..76f4314P. doi:10.1103/PhysRevC.76.064314.
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.
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.

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[1] Repetowicz & Richmond (2005)

[2] Perez-Martin & Robledo (2007)

[3] Bartosch (2001)

[4] Michalowicz et al. (2009)

[5] Isserlis (1918)

[6] Isserlis (1916)

Isserlis' theorem

Theorem statement

The Isserlis theorem

See also

References