On an estimate of the correlation function of a homogeneous and isotropic Gaussian random field. (Russian) Zbl 0577.60049

Dykhovichnyj, A. A.

Teor. Veroyatn. Mat. Stat. 29, 37-40 (1983).

A homogeneous, isotropic and in the quadratic mean continuous Gaussian field \(\xi\) in \(R^ n\) is considered. It is assumed that the field is observed on a sphere with radius R. The correlation function B(x,y) of \(\xi\) depends only on the distance r of x and y, \(B(x,y)=B(r)\). An estimate \(\hat B(\)r) for B(r) is introduced. Under some boundedness and continuity conditions for the spectral function of \(\xi\) it is shown, that \(\lim_{R\to \infty}(R-r)^ nD(\hat B(r))\) is proportional to a function of r which is closely connected with a representation of B(r).

Reviewer: C.Baldauf

MSC:

60G60

Random fields

Keywords:

homogeneous and isotropic Gaussian random field; continuity conditions for the spectral function