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

• 成果类型: Article
• 署名作者: Lavancier, Frederic; Philippe, Anne
• 署名单位: Nantes Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0262-2
• 发表日期: 2011
• 页码: 493-514
• 关键词: CENTRAL LIMIT-THEOREMS; random-variables

摘要

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.