## Limit theorems for characteristics of a crossing of a low level by Gaussian random fields.(English. Russian original)Zbl 0842.60051

Theory Probab. Math. Stat. 47, 91-100 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 88-99 (1992).
The authors discuss the functionals $W_1 (r)= \int_{V(r)} \max \{0, b(r)- \xi(x) \}dx, \qquad W_2 (r)= \int_{V(r)} \max \{0, b(r)- |\xi (x)|\}dx$ generated by a real measurable separable mean-square continuous homogeneous isotropic Gaussian random field $$\xi (x)$$, $$x\in \mathbb{R}^n$$, with $$E\xi (x)=0$$, $$E\xi^2 (x)=1$$ and correlation function $$B(|x|)= L(|x|) |x|^{-\alpha}$$, $$\alpha>0$$. Here $$V(r)= \{x\in \mathbb{R}^n: |x|< r\}$$, $$b(r)$$ is a positive nonincreasing function such that $$b(r)\to 0$$ as $$r\to 0$$ and $$L(\cdot)$$ is a nonnegative function slowly varying at infinity. It is proved that under some additional assumptions on $$L(\cdot)$$ and $$B(\cdot)$$ properly normalized variables $$(W_1 (r)- EW_1 (r))/ a_1 (r)$$ are asymptotically $${\mathcal N} (0,1)$$-distributed and limit distributions of $$(W_2 (r)- EW_2 (r))/ a_2 (r)$$ and $$Y_2 (r)=- \int_{V(r)} (\xi^2 (x)- 1)dx/ \delta_2 (r)$$ are equal (provided that one of them exists).

### MSC:

 60G60 Random fields 60G15 Gaussian processes 60G18 Self-similar stochastic processes 60F05 Central limit and other weak theorems