## A precise estimate of the rate of convergence in central limit theorem in Hilbert space.(Russian)Zbl 0694.60004

The paper was already announced in a preprint, see the reviewer’s remark to S. V. Nagaev’s paper, Sib. Mat. Zh. 30, No.3, (175), 84-96 (1989; Zbl 0675.60010). Let $$\{X_ i$$, $$i\geq 1\}$$ be a sequence of i.i.d. r.v.’s in a separable Hilbert space H. Assume that E $$X_ 1=0$$, E $$| X_ 1|^ 2=\sigma^ 2$$ and E $$| X_ 1|^ 3=\beta <\infty$$. The main result is the optimal uniform Berry-Esséen type estimate: $(1)\quad | P(| n^{-1/2}\sigma^{-1}\sum_{i\leq n}X_ i-a| <x)-P(| Y-a| <x)| \leq C(\prod^{6}_{i=1}\sigma_ i^{-1})\sigma^ 3\beta (1+| a|^ 3)n^{-1/2},$ where C is an unknown absolute constant and $$a\in H$$, and Y denotes a suitable Gaussian r.v.
The bound (1) is in accordance with the asymptotically precise estimate due to the authors in J. Multivariate Anal. 28, No.2, 304-330 (1989; Zbl 0675.60011). In the Abstracts of Commun. of the 5th Int. Vilnius Conf. Probab. Theory Math. Statist., Vol. 2, 139-140 (1989), the authors announced a nonuniform bound, where instead of $$1+| a|^ 3$$ on the right hand side of (1) appears: $(1+\min (| a|^ 3,| x|^ 3))(1+| | a| -| x| |^ 3)^{-1}.$