Central limit theorems for sequences with \(m(n)\)-dependent main part
A triangular array \((X_ i^{(n)}, 1\leq i\leq h(n), n\in N)\) is called \(m(n)\)-dependent if for all \(n\in N\) and \(k\in\{2,\ldots,h(n)-m(n)\}\) the random vectors \((X_ i^{(n)},1\leq i\leq k-1)\) and \((X_ j^{(n)}, m(n)+k\leq j\leq h(n))\) are independent. A triangular array \((X_ i^{(n)})\) is said to have \(m(n)\)-dependent main part \((X_{i,m(n)})\) and a residual part \((\overline X_{i,m(n)})\) if \(X_ i^{(n)}=X_{i,m(n)}+\overline X_{i,m(n)}\) for all \(n\in N\), \(1\leq i\leq h(n)\), and \((X_{i,m(n)})\) is \(m(n)\)-dependent. Under various conditions the central limit theorem for such triangular arrays \((X_ i^{(n)})\) is considered. \textit{K. N. Berk}'s theorem [Ann. Probab. 1, 352-354 (1973; Zbl 0263.60006)] for \(m(n)\)-dependent random variables is used essentially.
scientific article; zbMATH DE number 95577
scientific article