The expected Euler characteristic approximation to excursion probabilities of Gaussian vector fields

TL;DR

This paper demonstrates that for smooth Gaussian vector fields, the joint excursion probability at high levels can be accurately approximated by the expected Euler characteristic of the excursion set, with super-exponentially small error.

Contribution

It verifies the expected Euler characteristic heuristic for a broad class of smooth Gaussian vector fields, extending previous results to joint excursion probabilities.

Findings

Approximation of joint excursion probabilities by expected Euler characteristic.

Error in approximation is super-exponentially small.

Validates the heuristic for a large class of Gaussian vector fields.

Abstract

Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion probability $\mathbb{P} \{\sup_{t\in T} X(t) \geq u, \sup_{s\in S} Y(s) \geq u \}$ can be approximated by $\mathbb{E}\{\chi(A_u)\}$, the expected Euler characteristic of the excursion set $A_u=\{(t,s)\in T\times S: X(t) \ge u, Y(s) \ge u\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic (cf. Taylor, Takemura and Alder (2005), Alder and Taylor (2007)) for a large class of smooth Gaussian vector fields.