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

Dan Cheng; Yimin Xiao

doi:10.1214/15-AAP1101

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

Dan Cheng, Yimin Xiao

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

Let X = {X(t), t ϵ R^N} be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ R^N and u ϵ R, denote by A_u = {t ϵ T :X(t) ≥ u} the excursion set. Under X(·) ∈ C²(R^N) and certain regularity conditions, the mean Euler characteristic of A_u, denoted by E{ø(A_u)}, 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(A_u)} such that the error is exponentially smaller than E{ø(A_u)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.

Original language	English (US)
Pages (from-to)	722-759
Number of pages	38
Journal	Annals of Applied Probability
Volume	26
Issue number	2
DOIs	https://doi.org/10.1214/15-AAP1101
State	Published - Apr 2016
Externally published	Yes

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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10.1214/15-AAP1101

Cite this

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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",

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AU - Cheng, Dan

AU - Xiao, Yimin

N1 - Publisher Copyright: © Institute of Mathematical Statistics, 2016.

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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.

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KW - Excursion probability

KW - Excursion set

KW - Gaussian random fields with stationary increments

KW - Super-exponentially small

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

Abstract

Keywords

ASJC Scopus subject areas

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