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: \[ (1+\min (| a|^ 3,| x|^ 3))(1+| | a| -| x| |^ 3)^{-1}. \]
Reviewer: L.Paditz


60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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