### Abstract

For a centered Gaussian random field X={X(t),t∈RN}, let _{T1} and _{T2} be two compact sets in RN such that I = _{T1} ∩ _{T2} ≠ 0{combining long solidus overlay} and denote by χ (_{A u} (I) ) the Euler characteristic of the excursion set _{A u} (I) = {t ∈ I: X (t) ≥ u} We show that under certain smoothness and regularity conditions, as u → ∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.

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

Pages (from-to) | 79-82 |

Number of pages | 4 |

Journal | Statistics and Probability Letters |

Volume | 92 |

DOIs | |

State | Published - Sep 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Euler characteristic
- Excursion probability
- Excursion set
- Gaussian random fields

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Double extreme on joint sets for Gaussian random fields.** / Cheng, Dan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Double extreme on joint sets for Gaussian random fields

AU - Cheng, Dan

PY - 2014/9

Y1 - 2014/9

N2 - For a centered Gaussian random field X={X(t),t∈RN}, let T1 and T2 be two compact sets in RN such that I = T1 ∩ T2 ≠ 0{combining long solidus overlay} and denote by χ (A u (I) ) the Euler characteristic of the excursion set A u (I) = {t ∈ I: X (t) ≥ u} We show that under certain smoothness and regularity conditions, as u → ∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.

AB - For a centered Gaussian random field X={X(t),t∈RN}, let T1 and T2 be two compact sets in RN such that I = T1 ∩ T2 ≠ 0{combining long solidus overlay} and denote by χ (A u (I) ) the Euler characteristic of the excursion set A u (I) = {t ∈ I: X (t) ≥ u} We show that under certain smoothness and regularity conditions, as u → ∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.

KW - Euler characteristic

KW - Excursion probability

KW - Excursion set

KW - Gaussian random fields

UR - http://www.scopus.com/inward/record.url?scp=84901328984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901328984&partnerID=8YFLogxK

U2 - 10.1016/j.spl.2014.05.001

DO - 10.1016/j.spl.2014.05.001

M3 - Article

AN - SCOPUS:84901328984

VL - 92

SP - 79

EP - 82

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

ER -