On the Slepian process of a random Gaussian trigonometricpolynomial

It is proved that the Slepian process of a stationary trigonometric polynomial with Gaussian coefficients has a Karhunen-Loeve expansion consisting of a finite number of terms, and that each eigenfunction is itself a finite trigonometric polynomial. Upper bounds for the error which results when replacing the Slepian process corresponding to a general Gaussian stationary process by the Slepian process corresponding to its finite trigonometric approximation are obtained. A numerical example is given and the results are used to estimate by simulation the distribution of the excursion time above a level of a particular Gaussian stationary process