Fourth mixed moments of Gaussian laws #
The fourth centered moment of four linear observables is the sum of the three products of
their pairwise covariances. The formula allows repeated observables and singular covariance.
For a multivariate Gaussian this gives the coordinate identity
E[Zᵢ Zⱼ Zₖ Zₗ] = Sᵢⱼ Sₖₗ + Sᵢₖ Sⱼₗ + Sᵢₗ Sⱼₖ for centered coordinates.
The linear-observable result is proved for Gaussian measures on real Banach spaces, then specialized to the multivariate law. Finite products of affine linear observables are integrable. Polarizing the existing scalar fourth-moment formula avoids any invertibility requirement.
References #
- L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12 (1918), 134–139.
Finite products of affine linear observables of a Gaussian measure are integrable, including the empty product.
The 2nth centered moment of a linear observable of a Gaussian measure is
Varⁿ (2n - 1)‼, with the variance expressed by the covariance bilinear form.
The fourth centered mixed moment of four linear observables of a Gaussian measure is the sum over their three pairings. No independence or nondegeneracy is assumed.
Every shifted coordinate monomial is integrable under every multivariate Gaussian.
Taking c = 0 gives raw monomials and taking c = m gives centered monomials.
No validity assumption on the covariance parameter is needed.
The fourth centered coordinate moment of a multivariate Gaussian, including repeated coordinates and positive-semidefinite singular covariance matrices.
Isserlis' fourth-moment identity for centered multivariate Gaussian coordinates.
The fourth centered coordinate moment of a random vector with a multivariate Gaussian law. Repeated coordinates and singular covariance matrices are allowed.