Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

TL;DR

This paper develops a method to construct triplewise independent sequences with arbitrary marginals, demonstrating that the classical CLT can fail in this setting, and explores conditions under which CLT holds for higher-order independence.

Contribution

The paper introduces a simple methodology to construct triplewise independent sequences with arbitrary marginals and analyzes their asymptotic behavior, highlighting cases where the CLT fails.

Findings

Classical CLT fails for certain triplewise independent sequences.

Constructed sequences have non-Gaussian asymptotic distributions.

For K ≥ 4, the sequences tend to satisfy the CLT, explained heuristically.

Abstract

We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution $F$ (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of $F$). This allows us to illustrate the extent of the 'failure' of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer $K$, new $K$-tuplewise independent sequences that are not mutually independent. For $K \geq 4$, it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.