### 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(·) ∈ C^{2}(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 | |

State | Published - Apr 2016 |

Externally published | Yes |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 26, no. 2, pp. 722-759. https://doi.org/10.1214/15-AAP1101

}

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 -