On the spectral density and asymptotic normality of weakly dependent random fields (Q1187529)

Let \(X=(X_ k,k\in\mathbb{Z}^ d)\) be a centered complex weakly stationary random fields. For any disjoint \({\mathcal S},{\mathcal D}\subset\mathbb{Z}^ d\) define \(\sup{EV\overline W\over(\| V\|_ 2\cdot\| W\|_ 2}=r(S,{\mathcal D})\) where the supremum is taken over all pairs of random variables \(V=\sum_{k\in S^*}a_ kX_ k\), \(W=\sum_{k\in{\mathcal D}^*}b_ kX_ k\), where \({\mathcal S}^*\) and \({\mathcal D}^*\) are finite subset of \({\mathcal S}\) and \({\mathcal D}\), and \(a_ k\) and \(b_ k\) are complex numbers. For every real number \(s\geq 1\), define \(r^*(s)=\sup r({\mathcal S},{\mathcal D})\), where the supremum is taken over all pairs of nonempty disjoint subsets \({\mathcal S},{\mathcal D}\subset\mathbb{Z}^ d:\text{dist}({\mathcal S},{\mathcal D})\geq s\). Theorem 1. If \(r^*(s)\to 0\) as \(s\to\infty\), then \(X\) has a continuous spectral density. Theorem 2. The following two statements are equivalent: (1) \(r^*(1)<1\) and \(r^*(s)\to 0\) as \(s\to\infty\); (2) \(X\) has a continuous spectral density. The central limit theorem is proved. No mixing rate is assumed.