Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field
type: Item
instance of: scholarly article
title: Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field
zbMATH Open document ID: 0753.60048
DOI: 10.1007/BF01066693
published in: Ukrainian Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(\xi(x)\), \(x\in R^ n\), be a homogeneous isotropic random field with \(E\xi(x)=0\), \(E\xi^ 2(x)=1\) and correlation function \[ B_ n(r)=\int^ \infty_ 0Y_ n(r\lambda)d\varphi(\lambda),\quad Y_ n(z)=2^{(n-2)/2}\Gamma(n/2)J_{(n-2)/2}(z)z^{(n-2)/2} \] such that \[ \int^ \infty_ 0z^{n-1}| B_ n(z)| dz=+\infty. \] If \(0<\alpha<n\), then the following statements are equivalent a) \(\varphi(\lambda)/\lambda^ \alpha\sim A\), \(\lambda\to+0\); b) \(b_ n(r)/r^{2n-\alpha}\sim c(n,\alpha)A\), \(r\to\infty\), where \(A\), \(c(n,\alpha)\) are constants; \(v(r)=\{x\in R^ n:| x| <r\}\), \[ b_ n(r)=\int_{v(r)}\int_{v(r)}B_ n(| x-y|)dx dy=\text{var}\left[\int_{v(r)}\xi(x)dx\right]. \]
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 62M40
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 58677
zbMATH Keywords: Tauberian theorem
zbMATH Keywords: Abelian theorems
zbMATH Keywords: homogeneous isotropic random field
zbMATH Keywords: correlation function
author: Nikolai N. Leonenko
author: Andriy Olenko
reviewed by: Nikolai N. Leonenko
MaRDI profile type: Publication
cites work: Q3883244
cites work: Q4693159
cites work: A Tauberian Theorem for Integral Transforms of Hankel Type
cites work: Q3772977
cites work: Q3957713
cites work: Q3732621
cites work: Expansion in series of homogeneous temperature functions of the first and second kinds
cites work: A Table of Ramanujan's Function τ(n )
cites work: COMPLEX FOURIER--BESSEL TRANSFORMS
cites work: Q5507915
cites work: Non-central limit theorems for non-linear functional of Gaussian fields
cites work: Convergence of integrated processes of arbitrary Hermite rank
cites work: Multiple Wiener-Ito integrals. With applications to limit theorems
description: scientific article; zbMATH DE number 58677