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Normal approximation for sums of non-identically distributed random variables in Hilbert spaces. (English) Zbl 0617.60022

The author proves that the error of approximation in the central limit theorem in Hilbert spaces for ellipsoidal regions and independent not necessarily identically distributed random vectors is of order \(O(n^{- 1/2})(1+\| a\|^ 3)\) where a is the center of the region. The explicit error bound is proved under the assumption that the covariance operators of the vectors commute and that 13 eigenvalues of the covariance of the sums are nonzero. The proofs rely on methods introduced by Yurinskij, Bentkus and the reviewer.
Reviewer: F.Götze

MSC:

60F05 Central limit and other weak theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
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