A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables

Abstract

By the Lindeberg-L\'evy central limit theorem, standardized partial sums of a sequence of mutually independent and identically distributed random variables converge in law to the standard normal distribution. It is known that mutual independence cannot be relaxed to pairwise and even not triplewise independence. Counterexamples have been constructed for most marginal distributions: a recent construction works under a condition which excludes certain probability distributions with atomic parts, in particular discrete distributions in the `general position.' In the present paper, we show that this condition can be lifted: for any probability distribution $ F $ on the real line, which has finite variance and is not concentrated in a single point, there exists a sequence of triplewise independent random variables with distribution $ F $, such that its standardized partial sums converge in law to a distribution which is not normal. There is also scope for extension to $ k $-tuplewise independence.