The expected Euler characteristic approximation to excursion probabilities of smooth Gaussian random fields with general variance functions

TL;DR

This paper proves that for smooth Gaussian fields with varying variance, the probability of high excursions can be accurately approximated by the expected Euler characteristic of the excursion set, with errors diminishing rapidly.

Contribution

It extends the expected Euler characteristic heuristic to Gaussian fields with general variance functions and provides explicit approximation formulas using the Laplace method.

Findings

Super-exponential decay of approximation error.

Validates the Euler characteristic heuristic for diverse Gaussian fields.

Provides explicit formulas for excursion probability approximations.

Abstract

Consider a centered smooth Gaussian random field $\{X(t), t\in T \}$ with a general (nonconstant) variance function. In this work, we demonstrate that as $u \to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t) \geq u\}$ can be accurately approximated by $\mathbb{E}\{\chi(A_u)\}$ such that the error decays at a super-exponential rate. Here, $A_u = \{t\in T: X(t)\geq u\}$ represents the excursion set above $u$, and $\mathbb{E}\{\chi(A_u)\}$ is the expectation of its Euler characteristic $\chi(A_u)$. This result substantiates the expected Euler characteristic heuristic for a broad class of smooth Gaussian random fields with diverse covariance structures. In addition, we employ the Laplace method to derive explicit approximations to the excursion probabilities.