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On the spectral density and asymptotic normality of weakly dependent random fields. (English) Zbl 0787.60059

Let $X=\left({X}_{k},k\in {ℤ}^{d}\right)$ be a centered complex weakly stationary random fields. For any disjoint $𝒮,𝒟\subset {ℤ}^{d}$ define $sup\frac{EV\overline{W}}{\left(\parallel V{\parallel }_{2}·\parallel W{\parallel }_{2}}=r\left(S,𝒟\right)$ where the supremum is taken over all pairs of random variables $V={\sum }_{k\in {S}^{*}}{a}_{k}{X}_{k}$, $W={\sum }_{k\in {𝒟}^{*}}{b}_{k}{X}_{k}$, where ${𝒮}^{*}$ and ${𝒟}^{*}$ are finite subset of $𝒮$ and $𝒟$, and ${a}_{k}$ and ${b}_{k}$ are complex numbers. For every real number $s\ge 1$, define ${r}^{*}\left(s\right)=supr\left(𝒮,𝒟\right)$, where the supremum is taken over all pairs of nonempty disjoint subsets $𝒮,𝒟\subset {ℤ}^{d}:\text{dist}\left(𝒮,𝒟\right)\ge s$.

Theorem 1. If ${r}^{*}\left(s\right)\to 0$ as $s\to \infty$, then $X$ has a continuous spectral density.

Theorem 2. The following two statements are equivalent: (1) ${r}^{*}\left(1\right)<1$ and ${r}^{*}\left(s\right)\to 0$ as $s\to \infty$; (2) $X$ has a continuous spectral density.

The central limit theorem is proved. No mixing rate is assumed.

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