Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models

Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define Ru(t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (Ru)u>0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.