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- Title
Extremes of Gaussian Processes with Maximal Variance near the Boundary Points.
- Authors
Hashorva, Enkelejd; Hüsler, Jürg
- Abstract
Let X(t), t€[0, 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{supt€[0, 1] X(t) > u } is known for u → infin;. We investigate the case where the unique point t0 = tu depends on M and tends to the boundary. This is reasonable for a family of Gaussian processes Xu (t) depending on u, which have for each u such a unique point tu tending to the boundary as u → ∞. We derive the asymptotic behavior of P{supt€[0, 1] X(t) > u }, depending on the rate as tu tends to 0 or 1. Some applications are mentioned and the computation of a panicular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeckprocess. It shows the difficulties to simulate such rare events, when u is large.
- Subjects
GAUSSIAN processes; DISTRIBUTION (Probability theory); STOCHASTIC processes; BEHAVIOR; BROWNIAN bridges (Mathematics)
- Publication
Methodology & Computing in Applied Probability, 2000, Vol 2, Issue 3, p255
- ISSN
1387-5841
- Publication type
Article
- DOI
10.1023/A:1010029228490