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A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

Avanzi, Benjamin and Boglioni Beaulieu, Guillaume and Lafaye de Micheaux, Pierre and Ouimet, Frédéric and Wong, Bernard (2021) A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables. Journal of Mathematical Analysis and Applictions, 499 (1). Art. No. 124982. ISSN 0022-247X. https://resolver.caltech.edu/CaltechAUTHORS:20210126-140117834

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Abstract

The classical Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of n mutually independent and identically distributed random variables with finite second moment converges in distribution to a standard Gaussian as n goes to infinity. In particular, pairwise independence of the sequence is generally not sufficient for the theorem to hold. We construct explicitly such a sequence of pairwise independent random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions) and for which no CLT holds. We obtain, in closed form, the asymptotic distribution of the sample mean of our sequence, and find it is asymmetrical for any F. This is illustrated through several theoretical examples for various choices of F. Associated R codes are provided in a supplementary appendix online.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jmaa.2021.124982DOIArticle
https://arxiv.org/abs/2003.01350arXivDiscussion Paper
ORCID:
AuthorORCID
Avanzi, Benjamin0000-0002-5424-4292
Boglioni Beaulieu, Guillaume0000-0003-0231-6191
Lafaye de Micheaux, Pierre0000-0002-0247-5136
Ouimet, Frédéric0000-0001-7933-5265
Wong, Bernard0000-0002-7124-5342
Alternate Title:A counterexample to the central limit theorem for pairwise independent random variables having a common absolutely continuous arbitrary margin
Additional Information:© 2021 Elsevier Inc. Received 18 May 2020, Available online 23 January 2021. B. A. and B. W. are supported by Australian Research Council's Linkage (LP130100723) and Discovery (DP200101859) Projects funding schemes. 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). The authors have no conflict of interest to disclose.
Funders:
Funding AgencyGrant Number
Australian Research CouncilLP130100723
Australian Research CouncilDP200101859
University of SydneyUNSPECIFIED
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, characteristic function, mutual independence, non-Gaussian asymptotic distribution, pairwise independence
Issue or Number:1
Classification Code:2020 MSC: Primary: 62E20; Secondary: 60F05, 60E10
Record Number:CaltechAUTHORS:20210126-140117834
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210126-140117834
Official Citation:Benjamin Avanzi, Guillaume Boglioni Beaulieu, Pierre Lafaye de Micheaux, Frédéric Ouimet, Bernard Wong, A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables, Journal of Mathematical Analysis and Applications, Volume 499, Issue 1, 2021, 124982, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2021.124982.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107731
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Jan 2021 22:15
Last Modified:04 Feb 2021 19:05

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