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Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin. (English) Zbl 1476.60047

Summary: We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution \(F\) (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of \(F)\). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer \(K\), new \(K\)-tuplewise independent sequences that are not mutually independent. For \(K \geq 4\), it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

MSC:

60F05 Central limit and other weak theorems
60E10 Characteristic functions; other transforms
62E20 Asymptotic distribution theory in statistics
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