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On the spectral density and asymptotic normality of weakly dependent random fields. (English) Zbl 0787.60059
Let $X=(X\sb k,k\in\bbfZ\sp d)$ be a centered complex weakly stationary random fields. For any disjoint ${\cal S},{\cal D}\subset\bbfZ\sp d$ define $\sup{EV\overline W\over(\Vert V\Vert\sb 2\cdot\Vert W\Vert\sb 2}=r(S,{\cal D})$ where the supremum is taken over all pairs of random variables $V=\sum\sb{k\in S\sp*}a\sb kX\sb k$, $W=\sum\sb{k\in{\cal D}\sp*}b\sb kX\sb k$, where ${\cal S}\sp*$ and ${\cal D}\sp*$ are finite subset of ${\cal S}$ and ${\cal D}$, and $a\sb k$ and $b\sb k$ are complex numbers. For every real number $s\ge 1$, define $r\sp*(s)=\sup r({\cal S},{\cal D})$, where the supremum is taken over all pairs of nonempty disjoint subsets ${\cal S},{\cal D}\subset\bbfZ\sp d:\text{dist}({\cal S},{\cal D})\ge s$. Theorem 1. If $r\sp*(s)\to 0$ as $s\to\infty$, then $X$ has a continuous spectral density. Theorem 2. The following two statements are equivalent: (1) $r\sp*(1)<1$ and $r\sp*(s)\to 0$ as $s\to\infty$; (2) $X$ has a continuous spectral density. The central limit theorem is proved. No mixing rate is assumed.

60G60Random fields
60G10Stationary processes
60F05Central limit and other weak theorems
Full Text: DOI
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