On the rate of convergence to the normal law for functionals of a vector Gaussian random field. (English. Ukrainian original) Zbl 0955.60057

Derijeva, O. M.

Theory Probab. Math. Stat. 60, 29-34 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 26-30 (1999).

Let \(\xi(x)=(\xi_1(x),\ldots,\xi_{p}(x))'\) be a separable mean-square continuous homogeneous isotropic random field with a known correlation matrix \(R_{\xi}(x)= E\xi(0)\xi^{'}(x)=(R_{ij}(\|x\|))_{i,j=1}^{p},\) where \(R_{ii}(\|x\|)= a(\|x\|)\), \(R_{ij}(\|x\|)=b(\|x\|)\), \(i\neq j\), \(a(\|x\|)=L(\|x\|)\|x\|^{-\gamma}\), \(b(\|x\|)=\rho L(\|x\|)\|x\|^{-\gamma}\), \(\gamma>0\), \(0\leq\rho<1,\) and \(L(t)\), \(t>0\), is a function bounded on any interval. Let \(T=( T_{ij})_{i,j=1}^{p}\) be a matrix and let \(G(y)\) be a function. Let \(\widetilde{G}(y)= G(T^{-1}y)\) and \(\eta(x)=T\xi(x).\) The author considers the functional \[ S_{r}=\Biggl(\int_{v(r)}\widetilde{G}(\eta(x)) dx- C(0)|v(r)|\Biggr)\Biggl(\sum_{|\alpha|\leq 0} C^2(\alpha)\sigma_{\alpha}^2(r)\Biggr)^{-1/2}, \] where \(C(\alpha)\) and \(\sigma_{\alpha}(r)\) are known constants, and \(v(r)=\{ x\in R^{n}, \|x\|\leq r\}.\) Let \(\Delta_{r}=\sup_t|P\{ S_{r}<t\}-\Phi(t)|\) be a homogeneous metric for distribution functions. Under some assumption the author finds estimates for \(\Delta_{r}\).

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G60

Random fields

Keywords:

Gaussian random field; normal law; convergence; Chebyshev-Hermite polynomials; isotropic fields; strong dependence