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

Beaulieu Guillaume Boglioni (), Lafaye de Micheaux Pierre () and Ouimet Frédéric ()
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Beaulieu Guillaume Boglioni: UNSW Sydney, NSW 2052, Australia.
Lafaye de Micheaux Pierre: 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.

Dependence Modeling, 2021, vol. 9, issue 1, 424-438

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.

Keywords: central limit theorem; graph theory; mutual independence; non-Gaussian asymptotic distribution; triplewise independence; variance-gamma distribution (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:9:y:2021:i:1:p:424-438:n:20

DOI: 10.1515/demo-2021-0120

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