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High-frequency asymptotics for subordinated stationary fields on an abelian compact group. (English) Zbl 1143.60007

Let \(G\) be a connected compact second countable abelian group, \(\widehat G\) be its character group, \(T=\{T(g):g\in G\}\) be a real valued stationary Gaussian field. Put \(F[T](g)=F(T(g)), \;g\in G\), where \(F\in L^2(\mathbb{R}, \exp({-x^2}/2)dx)\), and \(\widetilde a_\chi(F)=\int_G F[T](g)\chi(g^{-1})dg, \;\chi\in \widehat G\). The aim of the article is to establish sufficient (and in many cases, also necessary) conditions when the following CLT holds:
\[ \mathbb{E}[| \tilde a_\chi(F)| ^2]^{1/2}\tilde a_\chi(F)\overset{law}{\longrightarrow} N+iN', \]
where \(N\) and \(N'\) are two independent centered Gaussian random variables with common variance equal to \(1/2\). The convergence means that an infinite sequence elements \(\{\chi_l\}\) of \(\widehat G\) is taken and the limit is taken as \(l\rightarrow +\infty\). The proofs of the main results involved recently established criteria for the weak convergence of multiple Winer–Itô integrals.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F05 Central limit and other weak theorems
60G60 Random fields
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