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

Dan Cheng, Yimin Xiao

Research output: Contribution to journalArticle

11 Citations (Scopus)

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

Fingerprint

Gaussian Random Field
Excursion
Euler Characteristic
Increment
Regularity Conditions
Rectangle
Heuristics
Verify
Denote
Random field
Rice
Class
Regularity

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

Cite this

The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments. / Cheng, Dan; Xiao, Yimin.

In: Annals of Applied Probability, Vol. 26, No. 2, 04.2016, p. 722-759.

Research output: Contribution to journalArticle

@article{b7e79834b8424a89b86b18de6c37bae3,
title = "The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments",
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{{\o}(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{{\o}(Au)} such that the error is exponentially smaller than E{{\o}(Au)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.",
keywords = "Euler characteristic, Excursion probability, Excursion set, Gaussian random fields with stationary increments, Super-exponentially small",
author = "Dan Cheng and Yimin Xiao",
year = "2016",
month = "4",
doi = "10.1214/15-AAP1101",
language = "English (US)",
volume = "26",
pages = "722--759",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "2",

}

TY - JOUR

T1 - The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

AU - Cheng, Dan

AU - Xiao, Yimin

PY - 2016/4

Y1 - 2016/4

N2 - 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.

AB - 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.

KW - Euler characteristic

KW - Excursion probability

KW - Excursion set

KW - Gaussian random fields with stationary increments

KW - Super-exponentially small

UR - http://www.scopus.com/inward/record.url?scp=84964325628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964325628&partnerID=8YFLogxK

U2 - 10.1214/15-AAP1101

DO - 10.1214/15-AAP1101

M3 - Article

AN - SCOPUS:84964325628

VL - 26

SP - 722

EP - 759

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 2

ER -