Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Let X(t), t0,1], 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 fixed point t₀ in the interval 0,1], the behavior of P{sup_t0,1] X(t)>u} is known for u. We investigate the case where the unique point t₀ = 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 P{sup_t0,1] X(t)>u}, 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 difficulties to simulate such rare events, when u is large.