Janson, Svante Some pairwise independent sequences for which the central limit theorem fails. (English) Zbl 0645.60027 Stochastics 23, No. 4, 439-448 (1988). 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 Cited in 14 Documents MSC: 60F05 Central limit and other weak theorems Keywords:central limit theorem; counterexamples PDF BibTeX XML Cite \textit{S. Janson}, Stochastics 23, No. 4, 439--448 (1988; Zbl 0645.60027) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.