The excursion probability of Gaussian and asymptotically Gaussian random fields

The purpose of this thesis is to develop the asymptotic approximation to excursion probability of Gaussian and asymptotically Gaussian random fields. It is composed of two parts. The first part is to study smooth Gaussian random fields. We extend the expected Euler characteristic approximation to a wide class of smooth Gaussian random fields with non-constant variances. Applying similar techniques, we also find that the joint excursion probability of vector-valued smooth Gaussian random fields can be approximated via the expected Euler characteristic of related excursion sets. As useful applications, the excursion probabilities over random intervals and infinite intervals are also investigated. The second part focuses on non-smooth Gaussian and asymptotically Gaussian random fields. We study the excursion probability of Gaussian random fields on the sphere and obtain an asymptotics based on the Pickands' constant. Using double sum method, we also derive the approximation, which involves the generalized Pickands' constant, to excursion probability of anisotropic Gaussian and asymptotically Gaussian random fields.