On the strong law of large numbers for isotropic random fields. (Russian) Zbl 0566.60050

Gaposhkin, V. F.

Teor. Veroyatn. Mat. Stat. 30, 34-38 (1984).

Let \(\xi\) (x), \(x\in R^ n\), be an isotropic random field with \(E\xi (x)=0\), \(E\xi\) (x)\(\overline{\xi (y)}=E\xi (gx)\overline{\xi (gy)}\), \(g\in SO(n)\), and spectral representation \[ \xi (r,\theta_ 1,...,\theta_{n- 2},\tau)=\sum^{\infty}_{m=0}\sum^{h(n,m)}_{l=1}\xi^ l_ m(r)S^ l_ m(\theta_ 1,...,\theta_{n-2},\tau), \] where \(S^ l_ m\) are spherical harmonics and \(\xi^ l_ m(r)\) are uncorrelated random processes. Under some conditions on the correlation function of \(\xi^ 1_ 0(r):\) \(| v_ n(R)|^{-1}\int_{v_ n(R)}\xi (x)dx\to 0\) a.s., \(R\to \infty\), where \(v_ n(R)=\{| x| \leq R\}\).

Reviewer: N.N.Leonenko

MSC:

60G60	Random fields
60F15	Strong limit theorems

Keywords:

strong law of large numbers; isotropic random field; spectral representation; spherical harmonics; correlation function