A strictly stationary, N-tuplewise independent counterexample to the central limit theorem

TL;DR

This paper constructs a strictly stationary, N-tuplewise independent sequence of bounded random variables that defies the Central Limit Theorem, demonstrating its limitations under certain independence conditions.

Contribution

It provides the first explicit construction of such a sequence for any integer N ≥ 2, showing the CLT can fail under N-tuplewise independence.

Findings

The sequence is strictly stationary and N-tuplewise independent.

The CLT does not hold for this constructed sequence.

The construction extends previous non-stationary examples.

Abstract

For an arbitrary integer N that is at least 2, this paper gives a construction of a strictly stationary, N-tuplewise independent sequence of (non-degenerate) bounded random variables such that the Central Limit Theorem fails to hold. The sequence is in part an adaptation of a non-stationary example with similar properties constructed by one of the authors (ARP) in a paper published in 1998.