TY - JOUR

T1 - A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

AU - Avanzi, Benjamin

AU - Boglioni Beaulieu, Guillaume

AU - de Micheaux, Pierre Lafaye

AU - Ouimet, Frédéric

AU - Wong, Bernard

PY - 2021/7/1

Y1 - 2021/7/1

N2 - 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.

AB - 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.

KW - Central limit theorem

KW - Characteristic function

KW - Mutual independence

KW - Non-Gaussian asymptotic distribution

KW - Pairwise independence

UR - http://www.scopus.com/inward/record.url?scp=85100303288&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2021.124982

DO - 10.1016/j.jmaa.2021.124982

M3 - Article

AN - SCOPUS:85100303288

SN - 0022-247X

VL - 499

SP - 1

EP - 13

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

M1 - 124982

ER -