A strictly stationary, M-tuplewise independent counterexample to the Central Limit Theorem

TL;DR

This paper constructs a specific stationary sequence that is highly independent and mixing but still does not follow the Central Limit Theorem, challenging common assumptions in probability theory.

Contribution

It provides the first explicit example of a stationary, M-tuplewise independent, mixing sequence that violates the CLT, expanding understanding of dependence and limit theorems.

Findings

Sequence is stationary and M-tuplewise independent

Sequence is mixing in the ergodic sense

Sequence fails to satisfy the CLT

Abstract

For $M \geq 2$ we construct a strictly stationary, $M$-tuplewise independent sequence that is mixing (in the ergodic-theoretic sense) and yet still fails to satisfy the Central Limit Theorem.