A~precise estimate of the rate of convergence in the Central Limit Theorem in Hilbert~space

B. A. Zalesskii ; V. V. Sazonov ; V. V. Ulyanov

Sbornik. Mathematics, Tome 68 (1991) no. 2, pp. 453-482

Résumé

Let $$ S_n=n^{-1/2}\sigma^{-1}\sum_1^n(X_i-\mathbf EX_i),\quad\sigma^2=\mathbf E|X_1-\mathbf EX_1|^2, $$ be the normed sum of independent identically distributed random variables $X_i$ with values in a separable Hilbert space $H$. Denote by $V$ the covariance operator of $X$, and let $Y$ be an $H$-valued $(0,\sigma^{-2}V)$ Gaussian random variable. The authors prove that there exist an absolute constant such that for any $a\in H$ and $r\geqslant0$ $$ |\mathbf P(|S_n-a|)-\mathbf P(|Y-a|)|\leqslant c\biggl(\prod_1^6\sigma_i^{-1}\biggr)\sigma^3\mathbf E|X_1-\mathbf EX_1|^3(1+|a|^3)n^{-1/2}, $$ where $\sigma_1^2\geqslant\sigma_2^2\geqslant\dotsb$ are the eigenvalues of $V$. Up to the value of $c$, this estimate is unimprovable in general. Bibliography: 15 titles.