An Example on the Central Limit Theorem for Associated Sequences

Abstract

We construct a strictly stationary associated sequence $(X_n)_{n \in \mathbb{N}}$ with $EX_n = 0, 0 < EX^2_n < \infty$ such that $K(R) = \operatorname{Cov}(X_1, X_1) + \sum^R_{j=2} \operatorname{Cov}(X_1, X_j) \sim \log R$ as $R \rightarrow \infty$, but $\sum^n_{j=1} X_j/(nK(n))^{1/2}$ does not converge to $\mathscr{N}(0, 1)$ in distribution. This is a counterexample to a conjecture of Newman and Wright (1981).