The expected Euler characteristic approximation to excursion probabilities of smooth Gaussian random fields with general variance functions
Abstract
Consider a centered smooth Gaussian random field $\{X(t), t\in T \}$ with a general (nonconstant) variance function. In this work, we demonstrate that as $u \to \infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T} X(t) \geq u\}$ can be accurately approximated by $\mathbb{E}\{\chi(A_u)\}$ such that the error decays at a super-exponential rate. Here, $A_u = \{t\in T: X(t)\geq u\}$ represents the excursion set above $u$, and $\mathbb{E}\{\chi(A_u)\}$ is the expectation of its Euler characteristic $\chi(A_u)$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2023
- DOI:
- 10.48550/arXiv.2309.05627
- arXiv:
- arXiv:2309.05627
- Bibcode:
- 2023arXiv230905627C
- Keywords:
-
- Mathematics - Probability
- E-Print:
- arXiv admin note: text overlap with arXiv:2301.06634