Extremes of locally stationary Gaussian and chi fields on manifolds

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.