On the accuracy of Gaussian approximation for the probability of hitting a~ball

V. V. Yurinskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 270-278

Résumé

Let $X_1,X_2,\dots$ be independent random vectors in a separable Hilbert space $H$ such that $\mathbf EX_j=0$, $\mathbf E|X_j|^3\le L$ and $B$ is their common covariance operator. Let $Y$ be a centered Gaussian vector with a covariance operator $B/\operatorname{Sp} B$. Theorem 1. {\it For $a\in H$, $r\ge 0$ $$ |\mathbf P\{|a+S_n|\}-\mathbf P\{|a+Y|\}|\le cL(\operatorname{Sp}B)^{-1/2}(1+|a|^3)n^{-1/2}, $$ where $S_n=(X_1+\dots+X_n)(n\operatorname{Sp}B)^{-1/2}$ and $c$ depends on the spectrum of $B/\operatorname{Sp}B$ only.} The proof is based on the combination of results [2], [3].