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On an estimate of the rate of convergence in the central limit theorem in a Hilbert space. (English. Russian original) Zbl 0533.60029
Sov. Math., Dokl. 26, 575-578 (1982); translation from Dokl. Akad. Nauk SSSR 267, 276-279 (1982).
Let \(X_ j\), \(j=1,2,..\). be a sequence of i.i.d. random variables with values in a separable Hilbert space H. Assume that \(EX_ 1=0\) and \(E\| X_ 1\|^{3+p}<\infty\), \(0\leq p<1\). Let \(\gamma\) be an H- valued Gaussian random variable with mean 0 and the same covariance operator as \(X_ 1\). The author proves that for all \(r\geq 0\) \[ \sup \left| P \left( \| n^{-1/2} \sum^{n}_{j=1} X_ j \|< r \right) - P \left( \| \gamma \|<r \right) \right| = o\left(n^{-(1+p)/2}\right) \] as \(n\to\infty\). An example shows that this rate of convergence is typical only for balls about zero.
Reviewer: R.Wegmann
60F05 Central limit and other weak theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
41A25 Rate of convergence, degree of approximation
60G50 Sums of independent random variables; random walks