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Theory of Probability and Mathematical Statistics

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Bispectrum and a non-linear model for a non-Gaussian homogenous and isotropic field in 3D


Authors: György Terdik and László Nádai
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 95 (2016).
Journal: Theor. Probability and Math. Statist. 95 (2017), 153-172
MSC (2010): Primary 60G60, 62M15; Secondary 62M30
DOI: https://doi.org/10.1090/tpms/1027
Published electronically: February 28, 2018
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Abstract: The so-called bispectrum is a widely used construction for analyzing non-linear 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.


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Additional Information

György Terdik
Affiliation: Faculty of Informatics, University of Debrecen, Hungary
Email: terdik.gyorgy@inf.unideb.hu

László Nádai
Affiliation: John von Neumann Faculty of Informatics, Óbuda University, Budapest, Hungary
Email: nadai@uni-obuda.hu

DOI: https://doi.org/10.1090/tpms/1027
Keywords: Bispectrum, homogenous fields, isotropic fields, bicovariance, spherical Bessel-functions
Received by editor(s): October 31, 2016
Published electronically: February 28, 2018
Dedicated: Dedicated to Professor Nikolai N. Leonenko on the occasion of his 65th birthday
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society