A Family of Counterexamples to the Central Limit Theorem Based on Binary Linear Codes

Abstract

The central limit theorem (CLT) claims that the standardized sum of a random sequence converges in distribution to a normal random variable as the length tends to infinity. We prove the existence of a family of counterexamples to the CLT for d-tuplewise independent sequences of length n for all d=2,...,n-1. The proof is based on [n, k, d+1] binary linear codes. Our result implies that d-tuplewise independence is too weak to justify the CLT, even if the size d grows linearly in length n.