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

Keigo TAKEUCHI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E102-A   No.5   pp.738-740
Publication Date: 2019/05/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E102.A.738
Type of Manuscript: LETTER
Category: Coding Theory
Keyword: 
central limit theorem,  dependent random variables,  counterexamples,  binary linear codes,  

Full Text: FreePDF(231.7KB)

Summary: 
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.