Excursion Probability of Certain Non-centered Smooth Gaussian Random Fields

TL;DR

This paper demonstrates that for certain non-centered smooth Gaussian fields, the probability of high excursions can be accurately approximated by the expected Euler characteristic, with explicit formulas provided for specific cases.

Contribution

It verifies the Euler characteristic heuristic for non-centered Gaussian fields and offers more precise approximations than previous methods, including explicit formulas for key cases.

Findings

Excursion probability approximated by expected Euler characteristic with super-exponentially small error.

Explicit formulas derived for rectangular and spherical parameter spaces.

Improved accuracy over existing double sum methods.

Abstract

Let $X = \{X(t): t\in T \}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \{t\in T: X(t)\geq u\}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t)\ge u \}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\mathbb{E}\{\chi(A_u(X,T))\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic for a large class of non-centered smooth Gaussian random fields and provides a much more accurate approximation compared with those existing results by the double sum method. The explicit formulae for $\mathbb{E}\{\chi(A_u(X,T))\}$ are also derived for two cases: (i) $T$ is a rectangle and $X-\mathbb{E} X$ is stationary; (ii) $T$ is an $N$-dimensional sphere and $X-\mathbb{E} X$ is isotropic.