Extrema of locally stationary Gaussian fields on growing manifolds

Abstract

We consider a class of non-homogeneous, continuous, centered Gaussian random fields $\{X_h(t),t\in {\mathcal{M}}_h;0<h\le 1\}$ where ${\mathcal{M}}_h$ denotes a rescaled smooth manifold, i.e. ${\mathcal{M}}_h=\frac{1}{h}\mathcal{M},$ and study the limit behavior of the extreme values of these Gaussian random fields when $h$ tends to zero, which means that the manifold is growing. Our main result can be thought of as a generalization of a classical result of Bickel and Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).

28 pages, 1 figure

References in corpus (5)

• Generalized Mode and Ridge Estimation
• Density Level Sets: Asymptotics, Inference, and Visualization
• Theoretical Analysis of Nonparametric Filament Estimation
• Limit laws on extremes of non-homogeneous Gaussian random fields
• Excursion Probabilities of Isotropic and Locally Isotropic Gaussian Random Fields on Manifolds

Cited by in corpus (1)

• Extremes of Gaussian Random Fields with regularly varying dependence structure