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

Abstract

By the Lindeberg–Lévy 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 independence, nor even to 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 almost all distributions on a fixed finite set. 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.