Work

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

Year: 2024

Type: article

Abstract: Let {(X(t),Y(s)):t∈T,s∈S} be an R2-valued, centered, unit-variance smooth Gaussian vector field, where T and S are compact rectangles in the Euclidean space. It is shown that, as u→∞, the joint excursion probability P{supt∈TX(t)≥u,sups∈SY(s)≥u} can be approximated by E{χ(Au)}, the expected Euler characteristic of the excursion set Au={(t,s)∈T×S:X(t)≥u,Y(s)≥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.

Source: The Annals of Applied Probability

DOI: 10.1214/24-aap2102

Authors: Dan Cheng, Yimin Xiao

Institutions: Arizona State University, Michigan State University

Language: English

Topic: Point processes and geometric inequalities

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funder: National Science Foundation

Awards: Collaborative Research: Critical Points and Excursion Probability of Random Fields: Theory and Statistical Applications, ATD: Collaborative Research: A Geostatistical Framework for Spatiotemporal Extremes, Analysis and Geometry of Random Fields Related to Stochastic Partial Differential Equations and Random Matrices, Collaborative Research: Asymptotic Geometry and Analysis of Stochastic Partial Differential Equations