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

Boglioni Beaulieu, Guillaume and Lafaye de Micheaux, Pierre and Ouimet, Frédéric (2021) Counterexamples to the classical Central Limit Theorem for triplewise independent random variables having a common arbitrary margin. . (Unpublished)

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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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Boglioni Beaulieu, Guillaume0000-0003-0231-6191
Lafaye de Micheaux, Pierre0000-0002-0247-5136
Ouimet, Frédéric0000-0001-7933-5265
Additional Information:G. B. B. acknowledges financial support from UNSW Sydney under a University International Postgraduate Award, from UNSW Business School under a supplementary scholarship, and from the FRQNT (B2). F. O. is supported by a postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X supplement). This research includes computations using the computational cluster Katana supported by Research Technology Services at UNSW Sydney.
Funding AgencyGrant Number
University of New South WalesUNSPECIFIED
Fonds de recherche du Québec – Nature et technologies (FRQNT)UNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Subject Keywords:central limit theorem, graph theory, mutual independence, non-Gaussian asymptotic distribution, triplewise independence, variance-gamma distribution
Classification Code:2020 MSC: Primary: 62E20; Secondary: 60F05, 60E10
Record Number:CaltechAUTHORS:20210413-073459048
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108707
Deposited By: Tony Diaz
Deposited On:13 Apr 2021 21:42
Last Modified:19 Apr 2021 22:46

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