A strictly stationary, "causal," 5-tuplewise independent counterexample to the central limit theorem

TL;DR

This paper constructs a strictly stationary sequence of ±1 random variables with five-tuple independence and causality, which defies the central limit theorem by not converging to a normal distribution, challenging classical probabilistic assumptions.

Contribution

It introduces a novel counterexample sequence that is 5-tuplewise independent, causal, and stationary, yet violates the central limit theorem, expanding understanding of dependence structures.

Findings

Sequence is strictly stationary and 5-tuplewise independent.

Partial sums do not converge to a normal distribution.

Sequence has a trivial double tail sigma-field.

Abstract

A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent, (iii) the sequence is "causal" in a certain sense, (iv) the sequence has a trivial double tail sigma-field, and (v) regardless of the normalization used, the partial sums do not converge to a (nondegenerate) normal law. The example has some features in common with a recent construction (for an arbitrary fixed positive integer N), by Alexander Pruss and the author, of a strictly stationary N-tuplewise independent counterexample to the central limit theorem.