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On the spectral density and asymptotic normality of weakly dependent random fields. (English) Zbl 0787.60059
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.
Reviewer: N.Leonenko (Kiev)

##### MSC:
 60G60 Random fields 60G10 Stationary stochastic processes 60F05 Central limit and other weak theorems
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