Counterexamples to the classical Central Limit Theorem for triplewise independent random variables having a common arbitrary margin

Abstract

We construct explicitly two sequences of triplewise independent random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions) for which a Central Limit Theorem (CLT) does not hold. We obtain, in closed form, the asymptotic distributions of the sample means of those sequences, which are seen to depend on the specific choice of F. This allows us to illustrate the extent of the `failure' of the classical CLT under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent but dependent sequences (which are useful to assess the ability of independence tests to detect complex dependence). For K≥4, it appears that the sequences thus created do verify a CLT, and we explain heuristically why this is the case.