Theory of Probability and Mathematical Statistics
Bispectrum and a nonlinear model for non-Gaussian homogenous and isotropic field in 3D
Gy"orgy Terdik, L'aszl'o N'adai
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Abstract: The so-called bispectrum is a widely used construction for analyzing nonlinear time series. In this paper the generalized bispectrum of a homogenous and isotropic stochastic field in 3D is introduced. The isotropy is considered in third order, and we give some necessary and sufficient conditions for isotropy of homogenous random fields. The spatial three-point correlation function (bicovariance function) is given by the bispectrum in terms of a kernel function, which is a superposition of spherical Bessel-functions and Legendre-polynomials. In return, the same kernel function is used in expressing the bispectrum by the bicovariance function. As an example, we generalize a model for non-Gaussian fields, which is the sum of a Gaussian-field and its 2nd degree Hermite-polynomial. This model can be applied as an alternative to the Gaussian one used in Cosmology for non-Gaussian CMB temperature fluctuations.
Keywords: Bispectrum, homogenous fields, isotropic fields, bicovariance,spherical Bessel-functions
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