Asymptotic formula for the conjunction probability of smooth stationary Gaussian fields

TL;DR

This paper derives an asymptotic formula for the probability that multiple smooth stationary Gaussian fields simultaneously exceed a high threshold at some point, based on the geometry of their excursion sets.

Contribution

It provides a new asymptotic expression for conjunction probabilities of Gaussian fields, extending the understanding of high-level exceedances in multivariate Gaussian processes.

Findings

Asymptotic formula for conjunction probability as threshold u approaches infinity.

Connection between the probability and the volume of local maximum sets.

Partial validation of the Euler characteristic method for Gaussian fields.

Abstract

Let $\{X_i(t):\, t\in S\subset \R^d \}_{i=1,2,\ldots,n}$ be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as $\PP \left( \exists t\in S: \, X_i(t) \geq u, \, \forall i=1,2,\ldots,n \right)$ for a given threshold level $u$. As $u\rightarrow \infty$, we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Aza\"\i s and Wschebor describing the shape of the excursion set of a stationary centered Gaussian field. Our result confirms partially the validity of Euler characteristic method.