Extrema of rescaled locally stationary Gaussian fields on manifolds

Abstract

Given a class of centered Gaussian random fields $\{X_{h}(s),s\in\mathbb{R}^{n},h\in(0,1]\}$, define the rescaled fields $\{Z_{h}(t)=X_{h}(h^{-1}t),t\in\mathcal{M}\}$, where $\mathcal{M}$ is a compact Riemannian manifold. Under the assumption that the fields $Z_{h}(t)$ satisfy a local stationary condition, we study the limit behavior of the extreme values of these rescaled Gaussian random fields, as $h$ tends to zero. Our main result can be considered as a generalization of a classical result of Bickel and Rosenblatt (Ann. Statist. 1 (1973) 1071–1095), and also of results by Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367–379).