Open Access
May 2016 Excursion probability of Gaussian random fields on sphere
Dan Cheng, Yimin Xiao
Bernoulli 22(2): 1113-1130 (May 2016). DOI: 10.3150/14-BEJ688


Let $X=\{X(x)\colon\ x\in\mathbb{S}^{N}\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^{N}$. Approximations to the excursion probability $\mathbb{P}\{\sup_{x\in\mathbb{S}^{N}}X(x)\ge u\}$, as $u\to\infty$, 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 $\mathbb{P}\{\sup_{x\in\mathbb{S}^{N}}X(x)\ge u\}$ 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.


Download Citation

Dan Cheng. Yimin Xiao. "Excursion probability of Gaussian random fields on sphere." Bernoulli 22 (2) 1113 - 1130, May 2016.


Received: 1 January 2014; Revised: 1 June 2014; Published: May 2016
First available in Project Euclid: 9 November 2015

zbMATH: 1337.60102
MathSciNet: MR3449810
Digital Object Identifier: 10.3150/14-BEJ688

Keywords: Euler characteristic , excursion probability , Gaussian random fields on sphere , Pickands’ constant

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 2 • May 2016
Back to Top