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

2023/09/11 by Cheng, Dan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2309.05627

Abstract

Consider a centered smooth Gaussian random field \X(t), t∈ T \ with a general (nonconstant) variance function. In this work, we demonstrate that as u → ∞, the excursion probability ℙ\sup_{t∈ T} X(t) ≥ u\ can be accurately approximated by 𝔼\χ(A_u)\ such that the error decays at a super-exponential rate. Here, A_u = \t∈ T: X(t)≥ u\ represents the excursion set above u, and 𝔼\χ(A_u)\ is the expectation of its Euler characteristic χ(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.

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