### Abstract

Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let _{Au}(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability ℙ{_{supt∈T}X(t)≥u} can be approximated by the expected Euler characteristic of _{Au}(X,T), denoted by E{χ(_{Au}(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(_{Au}(X,T))} are also derived for two cases: (i) T is a rectangle and X-EX is stationary; (ii) T is an N-dimensional sphere and X-EX is isotropic.

Original language | English (US) |
---|---|

Pages (from-to) | 883-905 |

Number of pages | 23 |

Journal | Stochastic Processes and their Applications |

Volume | 126 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2016 |

Externally published | Yes |

### Fingerprint

### Keywords

- Euler characteristic
- Excursion probability
- Gaussian random fields
- Rectangle
- Sphere
- Super-exponentially small

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

**Excursion probability of certain non-centered smooth Gaussian random fields.** / Cheng, Dan.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 126, no. 3, pp. 883-905. https://doi.org/10.1016/j.spa.2015.10.003

}

TY - JOUR

T1 - Excursion probability of certain non-centered smooth Gaussian random fields

AU - Cheng, Dan

PY - 2016/3

Y1 - 2016/3

N2 - Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability ℙ{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T), denoted by E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))} are also derived for two cases: (i) T is a rectangle and X-EX is stationary; (ii) T is an N-dimensional sphere and X-EX is isotropic.

AB - Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability ℙ{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T), denoted by E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))} are also derived for two cases: (i) T is a rectangle and X-EX is stationary; (ii) T is an N-dimensional sphere and X-EX is isotropic.

KW - Euler characteristic

KW - Excursion probability

KW - Gaussian random fields

KW - Rectangle

KW - Sphere

KW - Super-exponentially small

UR - http://www.scopus.com/inward/record.url?scp=84957038586&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957038586&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2015.10.003

DO - 10.1016/j.spa.2015.10.003

M3 - Article

AN - SCOPUS:84957038586

VL - 126

SP - 883

EP - 905

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 3

ER -