The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Abstract

Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in \mathbb{R}$, denote by $A_u=\{t\in T:X(t)\ge u\}$ the excursion set. Under $X(\cdot )\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by $\mathbb{E}\{\phi (A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to \infty$, the excursion probability $\mathbb{P}\{{\sup }_{t\in T}X(t)\ge u\}$ can be approximated by $\mathbb{E}\{\phi (A_u)\}$ such that the error is exponentially smaller than $\mathbb{E}\{\phi (A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.

Published at http://dx.doi.org/10.1214/15-AAP1101 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Validity of the expected Euler characteristic heuristic
• On the distribution of the maximum of a gaussian field with d parameters
• On a class of space-time intrinsic random functions

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