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Excursion probability of Gaussian random fields on sphere. (English) Zbl 1337.60102
Authors’ 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 \(\mathbb{P} \{\sup {}_{x\in \mathbb{S}^{N}}X(x)\geq u\}\), as \(u\rightarrow \infty \), 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 [V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields. Translations of Mathematical Monographs 148. Providence, RI: AMS (1996; Zbl 0841.60024); T. L. Mikhaleva and V. I. Piterbarg, Theory Probab. Appl. 41, No. 2, 367–379 (1996); translation from Teor. Veroyatn. Primen. 41, No. 2, 438–451 (1996; Zbl 0883.60048); H. P. Chan and T. L. Lai, Ann. Probab. 34, No. 1, 80–121 (2006; Zbl 1106.60022)]. It is shown that the asymptotics of \( \mathbb{P} \{\sup {}_{x\in \mathbb{S}^{N}}\;X(x)\geq u\}\) are 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.

MSC:
60G60 Random fields
60G15 Gaussian processes
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