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BY 4.0 license Open Access Published by De Gruyter Open Access December 13, 2021

Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

  • Guillaume Boglioni Beaulieu , Pierre Lafaye de Micheaux and Frédéric Ouimet EMAIL logo
From the journal Dependence Modeling


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 [], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

MSC 2010: 62E20; 60F05; 60E10


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Received: 2021-08-03
Accepted: 2021-10-13
Published Online: 2021-12-13

© 2021 Guillaume Boglioni Beaulieu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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