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

Dan Cheng, Yimin Xiao

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

Let X = {X(t), t ϵ RN} be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ RN and u ϵ R, denote by Au = {t ϵ T :X(t) ≥ u} the excursion set. Under X(·) ∈ C2(RN) and certain regularity conditions, the mean Euler characteristic of Au, denoted by E{ø(Au)}, is derived. By applying the Rice method, it is shown that, as u→∞, the excursion probability P{suptϵT X(t) ≥ u} can be approximated by E{ø(Au)} such that the error is exponentially smaller than E{ø(Au)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.

Original languageEnglish (US)
Pages (from-to)722-759
Number of pages38
JournalAnnals of Applied Probability
Volume26
Issue number2
DOIs
StatePublished - Apr 2016
Externally publishedYes

Keywords

  • Euler characteristic
  • Excursion probability
  • Excursion set
  • Gaussian random fields with stationary increments
  • Super-exponentially small

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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