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

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 $$\Delta_{r}$$.

MSC:

 60G60 Random fields