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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
##### Keywords:
central limit theorem; counterexamples
Full Text:
##### 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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