Conditions for the spherical means of local functionals of Gaussian fields to converge to a Wiener process. (English. Russian original) Zbl 0633.60071

Leonenko, N. N.; Rybasov, K. V.

Theory Probab. Math. Stat. 34, 95-102 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 85-93 (1986).

Let \(\xi\) (x) be a homogeneous and isotropic random field, v(r) the ball of radius r and \(| v(r)|\) its volume. It is proved that under certain additional assumptions as \(r\to \infty\), the finite-dimensional distributions of the random processes \[ X_ r(t)=r^{- n/2}\{\int_{v(rt^{1/n})}J(\xi (x))dx-c_ 0r^ nt| v(1)| \},\quad t\in [0,1], \] converge to the finite-dimensional distributions of the Brownian motion process.

Reviewer: M.Yadrenko

MSC:

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

Keywords:

homogeneous and isotropic random field; finite-dimensional distributions; Brownian motion