On measures of exceeding a level by a Gaussian isotropic random field. (Russian) Zbl 0576.60043

Leonenko, N. N.

Teor. Veroyatn. Mat. Stat. 31, 64-82 (1984).

Let \(\xi (x)=\xi (\omega,x):\) \(\Omega \times R^ n\to R^ 1\) be a measurable, mean square continuous, almost surely continuous, homogeneous, isotropic Gaussian random field. Denote by v(r) the sphere of radius r in \(R^ n\). Let a and b be continuous monotonic functions on \(R^ 1_+\) such that \(\lim_{r\to \infty}a(r)=\lim_{r\to \infty}b(r)=\infty\). The author determines the asymptotic behaviour as \(r\to \infty\) of the functionals \(G_ a^{(1)}(\xi,r)=\lambda (\{x\in v(r): \xi (x)>a(r)\})\), \(G_ a^{(2)}(\xi,r)=\lambda (\{x\in v(r): | \xi (x)| >a(r)\})\), \(G_ b^{(3)}(\xi,r)=\lambda (\{x\in v(r): | \xi (x)| <b(r)\})\), and \(G^{(4)}_{a,b}(\xi,r)=G_ a^{(2)}(\xi,r)+G_ b^{(3)}(\xi,r)\). Here \(\lambda\) denotes Lebesgue measure.

Reviewer: M.Iosifescu

MSC:

60G60

Random fields

Keywords:

isotropic Gaussian random field; asymptotic behaviour