The Maximum of the Periodogram of a Non-Gaussian Sequence

It is a well-known fact that the periodogram ordinates of an lid mean-zero Gaussian sequence at the Fourier frequencies constitute an lid exponential vector, hence the maximum of these periodogram ordinates has a limiting Gumbel distribution. We show for a non-Gaussian lid mean-zero, finite variance sequence that this statement remains valid. We also prove that the point process constructed from the periodogram ordinates 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 lid finite variance sequence have very much the same asymptotic behavior as the same functionals applied to an lid exponential sample.