Abstract
Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (J. Contemp. Math. Anal. 34(2):1–15) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in L 2. Therefore the dichotomy observed in dimension d = 1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d > 1.
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Lavancier, F., Philippe, A. Some convergence results on quadratic forms for random fields and application to empirical covariances. Probab. Theory Relat. Fields 149, 493–514 (2011). https://doi.org/10.1007/s00440-010-0262-2
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DOI: https://doi.org/10.1007/s00440-010-0262-2
Mathematics Subject Classification (2000)
- 60F05
- 62M40
- 60F25
- 60G15
- 62M10
Keywords
- Gaussian random field
- Long memory
- Non central limit theorems
- Quadratic mean convergence
- Stochastic integral
- Triangular array