Limit distributions of spherical sojourn measures for vector Gaussian random fields. (English. Russian original) Zbl 0842.60050

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

Theory Probab. Math. Stat. 47, 83-90 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 80-88 (1992).

Let \(\xi(x)= [\xi_1 (x), \dots, \xi_p (x) ]^T\), \(x\in \mathbb{R}^n\), \(n\geq 2\), \(p\geq 2\), be a measurable separable mean-square continuous homogeneous isotropic vector Gaussian random field with strong dependence, \(S_n (r)\) be the sphere in \(\mathbb{R}^n\) of radius \(r>0\). The attention is concentrated on spherical sojourn measures, i.e. on functionals of the type \[ T_i (r)= l\{ x\in S_n (r): \xi(x)\in \Delta_i\}, \qquad i=1, 2, 4, \] where \(l(\cdot)\) is Lebesgue measure on \(S_n (r)\), \(\Delta_1\) is \(p\)-dimensional rectangle in \(\mathbb{R}^p\), \(\Delta_2\), \(\Delta_4\) are the balls in \(\mathbb{R}^p\) with constant or decreasing radius. Under some additional assumptions on correlation matrix of the field the asymptotic distributions of functionals \(T_i (r)\) as \(r\to \infty\) are obtained.

Reviewer: N.M.Zinchenko (Kiev)

MSC:

60G60	Random fields
60F05	Central limit and other weak theorems
60G15	Gaussian processes

Keywords:

Gaussian random field; Lebesgue measure; Bessel function; Hermite polynomials