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

Author

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  • Beaulieu Guillaume Boglioni

    (UNSW Sydney, NSW 2052, Australia.)

  • de Micheaux Pierre Lafaye

    (UNSW Sydney, NSW 2052, Australia; Desbrest Institute of Epidemiology and Public Health, Univ Montpellier, INSERM, Montpellier, France; AMIS, Université Paul Valéry Montpellier 3, France)

  • Ouimet Frédéric

    (McGill University, Montreal, QC, Canada.)

Abstract

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

Suggested Citation

  • Beaulieu Guillaume Boglioni & de Micheaux Pierre Lafaye & Ouimet Frédéric, 2021. "Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin," Dependence Modeling, De Gruyter, vol. 9(1), pages 424-438, January.
    • Handle: RePEc:vrs:demode:v:9:y:2021:i:1:p:424-438:n:20
    DOI: 10.1515/demo-2021-0120
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    References listed on IDEAS

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