## Bernoulli

• Bernoulli
• Volume 24, Number 3 (2018), 1834-1859.

### 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).

#### Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 1834-1859.

Dates
Revised: October 2016
First available in Project Euclid: 2 February 2018

https://projecteuclid.org/euclid.bj/1517540461

Digital Object Identifier
doi:10.3150/16-BEJ913

Mathematical Reviews number (MathSciNet)
MR3757516

Zentralblatt MATH identifier
06839253

#### Citation

Qiao, Wanli; Polonik, Wolfgang. Extrema of rescaled locally stationary Gaussian fields on manifolds. Bernoulli 24 (2018), no. 3, 1834--1859. doi:10.3150/16-BEJ913. https://projecteuclid.org/euclid.bj/1517540461

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