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Extrema of rescaled locally stationary Gaussian fields on manifolds. (English) Zbl 1429.60039

Summary: 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 P. J. Bickel and M. Rosenblatt [Ann. Stat. 1, 1071–1095 (1973; Zbl 0275.62033)], and also of results by T. L. Mikhaleva and V. I. Piterbarg [Theory Probab. Appl. 41, No. 2, 367–379 (1996; Zbl 0883.60048); translation from Teor. Veroyatn. Primen. 41, No. 2, 438–451 (1996)].

MSC:

60G15 Gaussian processes
60G60 Random fields
60G70 Extreme value theory; extremal stochastic processes
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