The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

TL;DR

This paper derives the mean Euler characteristic of excursion sets for Gaussian fields with stationary increments and shows it effectively approximates the excursion probability, confirming the heuristic in this context.

Contribution

It provides a rigorous derivation of the mean Euler characteristic for Gaussian fields with stationary increments and validates its use in approximating excursion probabilities.

Findings

Mean Euler characteristic derived under regularity conditions.

Excursion probability approximated by the Euler characteristic with exponentially small error.

Verifies the Euler characteristic heuristic for a broad class of Gaussian fields.

Abstract

Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}\{\varphi(A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by ${\mathbb{E}}\{\varphi(A_u)\}$ such that the error is exponentially smaller than ${\mathbb{E}}\{\varphi(A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.