Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field
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; \(v(r)=\{x\in R^ n:| x| <r\}\), \[ b_ n(r)=\int_{v(r)}\int_{v(r)}B_ n(| x-y|)dx dy=\text{var}\left[\int_{v(r)}\xi(x)dx\right]. \]
scientific article; zbMATH DE number 58677
scientific article