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.
Publié le : 1999-01-14
Classification:
Periodogram,
maximum,
order statistics,
point process,
Poisson process,
62M15,
60F05,
60G55,
60G70
@article{1022677270,
author = {Davis, Richard A. and Mikosch, Thomas},
title = {The Maximum of the Periodogram of a Non-Gaussian Sequence},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 522-536},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677270}
}
Davis, Richard A.; Mikosch, Thomas. The Maximum of the Periodogram of a Non-Gaussian Sequence. Ann. Probab., Tome 27 (1999) no. 1, pp. 522-536. http://gdmltest.u-ga.fr/item/1022677270/