Isserlis' theorem

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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 is a zero-mean multivariate normal random vector, then

where the sum is over all the pairings of , i.e. all distinct ways of partitioning into pairs , and the product is over the pairs contained in .[5][6]

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

Odd case, []

If is odd, there does not exist any pairing of . Under this hypothesis, Isserlis' theorem implies that

This also follows from the fact that has the same distribution as , which implies that .

Even case, []

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.

Generalizations[]

Gaussian integration by parts[]

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 , [9] and to prove the .[10]

Non-Gaussian random variables[]

For non-Gaussian random variables, the moment-cumulants formula[11] replaces the Wick's probability formula. If is a vector of random variables, then

where the sum is over all the partitions of , the product is over the blocks of and is the cumulant of .

See also[]

References[]

  1. ^ 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.
  2. ^ 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.
  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. arXiv:0707.3365. Bibcode:2007PhRvC..76f4314P. doi:10.1103/PhysRevC.76.064314. S2CID 119627477.
  4. ^ 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. S2CID 54500670.
  5. ^ Janson, Svante (June 1997). Gaussian Hilbert Spaces. Cambridge Core. 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. Bibcode:2009JSP...136...89M. doi:10.1007/s10955-009-9768-3. S2CID 119702133.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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. S2CID 55282482.
  10. ^ Remy, Guillaume (2020). "The Fyodorov–Bouchaud formula and Liouville conformal field theory". Duke Mathematical Journal. 169. arXiv:1710.06897. doi:10.1215/00127094-2019-0045. S2CID 54777103.
  11. ^ 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.

Further reading[]

  • Koopmans, Lambert G. (1974). The spectral analysis of time series. San Diego, CA: Academic Press.
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