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January 1999 The Maximum of the Periodogram of a Non-Gaussian Sequence
Richard A. Davis, Thomas Mikosch
Ann. Probab. 27(1): 522-536 (January 1999). DOI: 10.1214/aop/1022677270


It is a well-known fact that the periodogram ordinates of an iid mean-zero Gaussian sequence at the Fourier frequencies constitute an iid exponential vector, hence the maximum of these periodogram ordinates has a limiting Gumbel distribution. We show for a non-Gaussian iid mean-zero, finite variance sequence that this statement remains valid. We also prove that the point process constructed from the periodogram ordi-nates converges to a Poisson process. This implies the joint weak convergence of the upper order statistics of the periodogram ordinates. These results are in agreement with the empirically observed phenomenon that various functionals of the periodogram ordinates of an iid finite variance sequence have very much the same asymptotic behavior as the same functionals applied to an iid exponential sample.


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Richard A. Davis. Thomas Mikosch. "The Maximum of the Periodogram of a Non-Gaussian Sequence." Ann. Probab. 27 (1) 522 - 536, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 1073.62556
MathSciNet: MR1681157
Digital Object Identifier: 10.1214/aop/1022677270

Primary: 62M15
Secondary: 60F05 , 60G55 , 60G70

Keywords: Maximum , order statistics , periodogram , point process , Poisson process

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 1 • January 1999
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