Extremes of locally stationary Gaussian and chi fields on manifolds

TL;DR

This paper investigates the asymptotic behavior of extreme values of locally stationary Gaussian and chi fields on manifolds, considering fixed and shrinking parameter regimes, extending to chi fields.

Contribution

It provides new asymptotic results for the extremes of Gaussian and chi fields on manifolds, covering both fixed and vanishing parameter cases.

Findings

Asymptotic excursion probabilities for Gaussian fields on manifolds.

Limit laws for extremes of chi fields.

Extension of results to varying parameter regimes.

Abstract

Depending on a parameter $h\in (0,1]$, let $\{X_h(\mathbf{t})$, $\mathbf{t}\in\mathcal{M}_h\}$ be a class of centered Gaussian fields indexed by compact manifolds $\mathcal{M}_h$. For locally stationary Gaussian fields $X_h$, we study the asymptotic excursion probabilities of $X_h$ on $\mathcal{M}_h$. Two cases are considered: (i) $h$ is fixed and (ii) $h\rightarrow0$. These results are extended to obtain the limit behaviors of the extremes of locally stationary $\chi$-fields on manifolds.