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Central and non central limit theorems for quadratic forms of a strongly dependent Gaussian field. (English) Zbl 0881.60023

Summary: Strong dependence for a random field means that the autocorrelation function is not summable. In this case the usual central limit theorem for quadratic forms of the field does not necessarily hold. R. Fox and M. S. Taqqu [Ann. Probab. 13, 428-446 (1985; Zbl 0569.60016) and Probab. Theory Relat. Fields 74, 213-240 (1987; Zbl 0586.60019)] have studied the case of Gaussian processes. Using a representation of the quadratic form as a multiple Itô-Wiener stochastic integral, we extend these results to the case of Gaussian fields. In the one-dimensional case, our non central theorem is an extension of the result of Fox and Taqqu (1985).

MSC:

60F05 Central limit and other weak theorems
60G60 Random fields
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