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

Leonenko, N. N.; Sabirov, Sh. O.

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).

Reviewer: N.M.Zinchenko (Kiev)

MSC:

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

Keywords:

Gaussian random field; correlation function; Chebyshev-Hermite polynomials; limit distribution