Excursion probability of Gaussian random fields on sphere

Abstract

Let $X=\{X(x):x\in {\mathbb{S}}^N\}$ 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.

Published at http://dx.doi.org/10.3150/14-BEJ688 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere
• Critical radius and supremum of random spherical harmonics (II)
• Distribution of the Height of Local Maxima of Gaussian Random Fields
• Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields
• Tail Asymptotic Behavior of the supremum of a class of chi-square processes
• Isotropic covariance matrix functions on compact two-point homogeneous spaces