Excursion probability of Gaussian random fields on sphere

Dan Cheng, Yimin Xiao

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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 languageEnglish (US)
Pages (from-to)1113-1130
Number of pages18
Issue number2
StatePublished - May 2016
Externally publishedYes


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

ASJC Scopus subject areas

  • Statistics and Probability


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