A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables
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.
© 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.
Supplemental Material - 1-s2.0-S0022247X21000615-mmc1.pdf
Submitted - 2003.01350.pdf