Extremes of Gaussian Processes with Maximal Variance near the Boundary Points (original) (raw)

Let Xt, t[0Y 1, be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique ®xed point t 0 in the interval 0Y 1, the behavior of Pfsup t[0Y1 Xt4ug is known for u??. We investigate the case where the unique point t 0 t u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X u t depending on u, which have for each u such a unique point t u tending to the boundary as u??. We derive the asymptotic behavior of Pfsup t [ 0Y1 Xt4ug, depending on the rate as t u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the dif®culties to simulate such rare events, when u is large.