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

Guillaume Boglioni Beaulieu, Pierre Lafaye de Micheaux, Frédéric Ouimet*

*Corresponding author for this work

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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.

Original languageEnglish
Pages (from-to)424-438
Number of pages15
JournalDependence Modeling
Volume9
Issue number1
DOIs
Publication statusPublished - 13 Dec 2021
Externally publishedYes

Bibliographical note

Copyright © 2021 Guillaume Boglioni Beaulieu et al., published by De Gruyter. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Keywords

  • Central limit theorem
  • Graph theory
  • Mutual independence
  • Non-Gaussian asymptotic distribution
  • Triplewise independence
  • Variance-gamma distribution

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