Extrema of rescaled locally stationary Gaussian fields on manifolds

Given a class of centered Gaussian random fields {Xh(s), s ∈ R, h ∈ (0, 1]}, define the rescaled fields {Zh(t) = Xh(ht), t ∈ M}, where M is a compact Riemannian manifold. Under the assumption that the fields Zh(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 (1973a), and also of results by Mikhaleva and Piterbarg (1997). This research was partially support by the NSF-grant DMS 1107206 AMS 2000 subject classifications. Primary 60G70, 60G15.