• Bernoulli
  • Volume 22, Number 2 (2016), 1113-1130.

Excursion probability of Gaussian random fields on sphere

Dan Cheng and Yimin Xiao

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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.

Article information

Bernoulli, Volume 22, Number 2 (2016), 1113-1130.

Received: January 2014
Revised: June 2014
First available in Project Euclid: 9 November 2015

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


Cheng, Dan; Xiao, Yimin. Excursion probability of Gaussian random fields on sphere. Bernoulli 22 (2016), no. 2, 1113--1130. doi:10.3150/14-BEJ688.

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