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 Gum bel distribution. We show for a non-Gaussian lid mean-zero, finite variance sequence that this statement remains valid. We also prove that the point p rocess constructed from the periodogram ordinates converges to a Poisson pr ocess. This implies the joint weak convergence of the upper order statistic s of the periodogram ordinates. These results are in agreement with the emp irically observed phenomenon that various functionals of the periodogram or dinates of an lid finite variance sequence have very much the same asymptot ic behavior as the same functionals applied to an lid exponential sample.