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Some pairwise independent sequences for which the central limit theorem fails. (English) Zbl 0645.60027
The stationary sequence \(\{X_ k\}\) of pairwise independent complex r.v. is constructed in such a way that \(X_ k\) are uniformly bounded with mean 0 and finite non-zero variance \(\sigma^ 2\) and \(S_ n=\sum^{n}_{1}X_ i\to^{d}U(n\to \infty)\) for some non-normal r.v. U, while \(S_ n/(\sigma \sqrt{n})\to^{p}0\). Various modifications of the basic example are given and some problems are posed.
Reviewer: A.V.Bulinskii

MSC:
60F05 Central limit and other weak theorems
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References:
[1] Bioornfield P., Fourier Analysis of Time Series:An Introduction (1976)
[2] Bradley R., A stationary, pairwise independent, absoluteiy regular sequence for which the central limit theorem fails. Preprin (1987)
[3] Chung K. L., A Course in Probability Theory (1974)
[4] Grenander U., Probabilities on Algebraic Structures (1963) · Zbl 0131.34804
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