# Tauberian theorems for correlation functions and limit theorems for spherical averages of random fields

Tauberian theorems for correlation functions and limit theorems for spherical averages of random... Random Oper. & Stock. Equ., Vol. 1, No. 1, pp. 57-67 (1993) © VSP 1993 N. N. LEONENKO and A. Ya. OLENKO Department of Mechanics and Mathematics, Kyjiv University, 252017 Kyjiv, Ukraine Received for ROSE 15 February 1991 Abstract--Tauberian and Abelian theorems for integral transforms of Hankel type are proved. The limit theorems for spherical averages of functionals of homogeneous isotropic Gaussian random fields are considered. 1. INTRODUCTION Let Rn be an -dimensional Euclidean space, s(r) = {x G Rn: ||x|| = r} a sphere in R n , and vn(r) = {x G Rn: ||z|| < r} be a ball in R n . Let £(z), R n , be a real measurable square-mean continuous homogeneous isotropic Gaussian random field with E£(z) = 0, E£2(z) = 1, and the correlation function Bn(r) = Bn(\\x\\) = E£(0)£(z). It is known [1] that there exists a bounded nondecreasing function (), ^ 0, such that Bn(r) = 2<"- 2 > j J^(Ar)(Ar)< 2 -»>^(dA), (1) where 3v(z) is the i/th-order Bessel function of the first kind and It follows from the results of [1, 2] that = (2)2(-1> jL21(Ar)( r)2-^(dA) (2) (~3)/2 where ra(dx) is an element of Lebesgue measure http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Random Operators and Stochastic Equations de Gruyter

# Tauberian theorems for correlation functions and limit theorems for spherical averages of random fields

Random Operators and Stochastic Equations, Volume 1 (1) – Jan 1, 1993
12 pages

/lp/de-gruyter/tauberian-theorems-for-correlation-functions-and-limit-theorems-for-k8HHfSvqq0
Publisher
de Gruyter
ISSN
0926-6364
eISSN
1569-397X
DOI
10.1515/rose.1993.1.1.57
Publisher site
See Article on Publisher Site

### Abstract

Random Oper. & Stock. Equ., Vol. 1, No. 1, pp. 57-67 (1993) © VSP 1993 N. N. LEONENKO and A. Ya. OLENKO Department of Mechanics and Mathematics, Kyjiv University, 252017 Kyjiv, Ukraine Received for ROSE 15 February 1991 Abstract--Tauberian and Abelian theorems for integral transforms of Hankel type are proved. The limit theorems for spherical averages of functionals of homogeneous isotropic Gaussian random fields are considered. 1. INTRODUCTION Let Rn be an -dimensional Euclidean space, s(r) = {x G Rn: ||x|| = r} a sphere in R n , and vn(r) = {x G Rn: ||z|| < r} be a ball in R n . Let £(z), R n , be a real measurable square-mean continuous homogeneous isotropic Gaussian random field with E£(z) = 0, E£2(z) = 1, and the correlation function Bn(r) = Bn(\\x\\) = E£(0)£(z). It is known [1] that there exists a bounded nondecreasing function (), ^ 0, such that Bn(r) = 2<"- 2 > j J^(Ar)(Ar)< 2 -»>^(dA), (1) where 3v(z) is the i/th-order Bessel function of the first kind and It follows from the results of [1, 2] that = (2)2(-1> jL21(Ar)( r)2-^(dA) (2) (~3)/2 where ra(dx) is an element of Lebesgue measure

### Journal

Random Operators and Stochastic Equationsde Gruyter

Published: Jan 1, 1993

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