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, quantum field theory and generation of colored noise.
The Isserlis theorem
If (x1, …, x2n) is a zero mean multivariate normal random vector, then
where the notation ∑ ∏ means summing over all distinct ways of partitioning x1, …, x2n into pairs xi,xj and each summand is the product of the n pairs. This yields terms in the sum. 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).
For sixth-order moments, Isserlis' theorem is:
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- 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.
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- 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. 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. doi:10.1103/PhysRev.80.268.