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Some convergence results on quadratic forms for random fields and application to empirical covariances
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  • Published: 11 February 2010

Some convergence results on quadratic forms for random fields and application to empirical covariances

  • Frédéric Lavancier1 &
  • Anne Philippe1 

Probability Theory and Related Fields volume 149, pages 493–514 (2011)Cite this article

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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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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes, 44322, Nantes, France

    Frédéric Lavancier & Anne Philippe

Authors
  1. Frédéric Lavancier
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  2. Anne Philippe
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Correspondence to Anne Philippe.

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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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  • Received: 08 October 2009

  • Revised: 07 January 2010

  • Published: 11 February 2010

  • Issue Date: April 2011

  • DOI: https://doi.org/10.1007/s00440-010-0262-2

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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
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