### Abstract

Let X = {X(x): x e SN} be a real-valued, centered Gaussian random field indexed on the N-dimensional unit sphere SN. Approximations to the excursion probability P{supxeSN X(x) u}, as → ∞ are obtained for two cases: (i) X is locally isotropic and its sample functions are non-smooth and; (ii) X is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367-379) and Chan and Lai (Ann. Probab. 34 (2006) 80-121). It is shown that the asymptotics of P{supxeSN X(x) ≥ u} is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.

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

Pages (from-to) | 1113-1130 |

Number of pages | 18 |

Journal | Bernoulli |

Volume | 22 |

Issue number | 2 |

DOIs | |

State | Published - May 2016 |

Externally published | Yes |

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### Keywords

- Euler characteristic
- Excursion probability
- Gaussian random fields on sphere
- Pickands' constant

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Bernoulli*,

*22*(2), 1113-1130. https://doi.org/10.3150/14-BEJ688

**Excursion probability of Gaussian random fields on sphere.** / Cheng, Dan; Xiao, Yimin.

Research output: Contribution to journal › Article

*Bernoulli*, vol. 22, no. 2, pp. 1113-1130. https://doi.org/10.3150/14-BEJ688

}

TY - JOUR

T1 - Excursion probability of Gaussian random fields on sphere

AU - Cheng, Dan

AU - Xiao, Yimin

PY - 2016/5

Y1 - 2016/5

N2 - Let X = {X(x): x e SN} be a real-valued, centered Gaussian random field indexed on the N-dimensional unit sphere SN. Approximations to the excursion probability P{supxeSN X(x) u}, as → ∞ are obtained for two cases: (i) X is locally isotropic and its sample functions are non-smooth and; (ii) X is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367-379) and Chan and Lai (Ann. Probab. 34 (2006) 80-121). It is shown that the asymptotics of P{supxeSN X(x) ≥ u} is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.

AB - Let X = {X(x): x e SN} be a real-valued, centered Gaussian random field indexed on the N-dimensional unit sphere SN. Approximations to the excursion probability P{supxeSN X(x) u}, as → ∞ are obtained for two cases: (i) X is locally isotropic and its sample functions are non-smooth and; (ii) X is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367-379) and Chan and Lai (Ann. Probab. 34 (2006) 80-121). It is shown that the asymptotics of P{supxeSN X(x) ≥ u} is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.

KW - Euler characteristic

KW - Excursion probability

KW - Gaussian random fields on sphere

KW - Pickands' constant

UR - http://www.scopus.com/inward/record.url?scp=84958681837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958681837&partnerID=8YFLogxK

U2 - 10.3150/14-BEJ688

DO - 10.3150/14-BEJ688

M3 - Article

AN - SCOPUS:84958681837

VL - 22

SP - 1113

EP - 1130

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -