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Some convergence results on quadratic forms for random fields and application to empirical covariances. (English) Zbl 1234.60029
This paper considers the convergence in distribution of quadratic forms defined as the weighted sum of the cross-products from a sequence of \(d\)-dimensional stationary \(L^2\) Gaussian random fields having long memory. When \(d=1\), both central and non-central limit theorems have been proved under various conditions on the spectral density \(f\) of the sequence and weight function \(g\) defining the quadratic forms. When \(d>1\), a central limit theorem has been proved but few results are available on a non-central limit theorem. In this paper, a non-central limit theorem is provided for the convergence of the quadratic form under a general condition on \(f\) and \(g\) in the case \(d>1\). Some examples, where the general condition is satisfied, are given. As an application of the main result, the asymptotic distributions of an empirical auto-covariance function are derived under various conditions.

60F05 Central limit and other weak theorems
60G60 Random fields
60H40 White noise theory
60G10 Stationary stochastic processes
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