An Example on the Central Limit Theorem for Associated Sequences

Herrndorf, Norbert

Ann. Probab., Tome 12 (1984) no. 4, p. 912-917 / Harvested from Project Euclid

Résumé

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).

Publié le : 1984-08-14
Classification: Central limit theorem, strictly stationary associated sequences, 60F05

@article{1176993241,
     author = {Herrndorf, Norbert},
     title = {An Example on the Central Limit Theorem for Associated Sequences},
     journal = {Ann. Probab.},
     volume = {12},
     number = {4},
     year = {1984},
     pages = { 912-917},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176993241}
}

Herrndorf, Norbert. An Example on the Central Limit Theorem for Associated Sequences. Ann. Probab., Tome 12 (1984) no. 4, pp.  912-917. http://gdmltest.u-ga.fr/item/1176993241/