Asymptotic normality of a certain functional of geometric character of a Gaussian random field. (English. Russian original) Zbl 0746.60041

Leonenko, N. N.; Parkhomenko, V. N.

Theory Probab. Math. Stat. 42, 83-93 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 72-82 (1990).

Let \(\xi (x) = \xi (\omega ,x):\Omega\times\mathbb{R}^ 2 \to \mathbb{R}\) be a measurable separable homogeneous isotropic Gaussian random field with \(\text{E} \xi (x) = 0,\;\text{E}\xi^ 2 (x) = 1\) which possesses mean- square partial derivatives. By means of the Hermite polynomials technique and the method of moments conditions in terms of the spectral function of \(\xi\) are obtained under which \[ \bigl( S(r)-A(r)\bigr)/r\sqrt{c_ 2} {\overset\;{\mathcal D}\longrightarrow} N(0,1) \] as \(r\to \infty\). Here \[ S(r) = \iint_{x_ 1^ 2+x_ 2^ 2\leq r^ 2}\left( 1+\left( \partial\xi/\partial x_ 1\right) ^ 2 + \left( \partial\xi/ \partial x_ 2\right) ^ 2\right)^{1/2} dx_ 1 dx_ 2\quad (r>0), \] \(\text{E} S(r)=A(r)\), \(\operatorname{var}S(r)\sim c_ 2 r^ 2\quad (r\to \infty)\) and \(A(r)\), \(c_ 2\) are indicated explicitly.

Reviewer: A.V.Bulinskij (Moskva)

MSC:

60G15	Gaussian processes
60G60	Random fields
60D05	Geometric probability and stochastic geometry
60F05	Central limit and other weak theorems

Keywords:

integral functionals; central limit theorem; Gaussian random field; Hermite polynomial technique