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

Richard C. Bradley|ArXiv.org|Nov 15, 2009

Probability and Risk Models4 citations

TL;DR

This paper constructs a strictly stationary, 5-tuplewise independent sequence of ±1-valued random variables that is causal and has a trivial double tail σ-field, yet whose normalized partial sums do not converge to a normal distribution—providing the first finite-state, causal counterexample to the central limit theorem for 5-tuplewise independence. The construction uses a deterministic function of i.i.d. innovations to ensure the required dependence structure and mixing properties while violating the CLT under any normalization.

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.

Motivation & Objective

• To resolve the open question of whether a strictly stationary, finite-state, causal counterexample to the central limit theorem exists for 5-tuplewise independence.
• To construct a sequence that satisfies 5-tuplewise independence but fails the CLT under any normalization.
• To ensure the sequence is causal (Markovian) and has a trivial double tail σ-field, enhancing its structural and probabilistic clarity.
• To extend prior results on N-tuplewise independent counterexamples by incorporating causality and Bernoulli-like properties.

Proposed method

• Define the sequence $X_k$ as a measurable function $f(\eta_k, \eta_{k-1}, \dots)$ of i.i.d. innovations $\eta_k$, ensuring causality and stationarity.
• Use a recursive construction based on a finite-state Markov chain with memory to enforce 5-tuplewise independence while avoiding joint independence.
• Ensure the double tail σ-field is trivial by leveraging the asymptotic independence of far-apart blocks in the causal structure.
• Construct the sequence so that the sixth absolute moment of the normalized partial sums remains bounded away from the normal limit (15), violating the CLT.
• Verify that the sequence is strictly stationary and that all 5-tuples of distinct indices are independent via combinatorial and measure-theoretic arguments.
• Use a coupling argument with an auxiliary process $X^*$ to prove the triviality of the double tail σ-field by contradiction, relying on independence of distant blocks.

Experimental results

Research questions

• RQ1 Can a strictly stationary, finite-state, causal sequence that is 5-tuplewise independent but fails the CLT be constructed?
• RQ2 Does 5-tuplewise independence suffice to guarantee convergence to a normal distribution under any normalization?
• RQ3 Can such a counterexample be constructed with a trivial double tail σ-field and a causal structure?
• RQ4 Is there a structural obstruction to extending this construction to N ≥ 6 tuplewise independence with the same properties?
• RQ5 Can a finite-state, causal, N-tuplewise independent counterexample to the CLT exist for N ≥ 6?

Key findings

• The constructed sequence is strictly stationary and takes values in \{-1, +1\} with equal probability.
• Every five distinct random variables in the sequence are mutually independent, though higher-order dependencies exist.
• The sequence is causal: each $X_k$ is measurable with respect to the past $\sigma$-field generated by $\{\eta_j : j \leq k\}$, ensuring a Markovian structure.
• The double tail σ-field is trivial, meaning all tail events have probability 0 or 1.
• The limsup of the sixth absolute moment of $S_n / \sqrt{n}$ is strictly less than 15, violating the normal limit which would require convergence to 15.
• There exists an infinite subsequence $T \subset \mathbb{N}$ such that $S_n / \sqrt{n}$ converges in distribution to a nondegenerate, non-normal measure $\mu$, proving failure of the CLT.