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

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.

### MSC:

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