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.

A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

“…As a final comment, note that some authors have studied CLTs under K-tuplewise independence (for K > 2); see, e.g., Pruss [28], Bradley and Pruss [8], Bradley [7], Weakley [34], Takeuchi [32]. It would be interesting to generalize our construction in that direction.…”

“…Weakley [34] further extends this construction by allowing the X j 's to have any symmetrical distribution (with finite variance). Takeuchi [32] showed that even if K grows linearly with the sample size n, a CLT need not be valid.…”