Excursion Probabilities of Isotropic and Locally Isotropic Gaussian Random Fields on Manifolds

2015/04/29 by Dan Cheng, Cheng, Dan
Environmental Science · Mathematics · #FOS: Mathematics #Geometry and complex manifolds #Hydrology and Drought Analysis #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1504.08047

openalex publication_date 2015/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X= \X(p), p∈ M\ be a centered Gaussian random field, where M is a smooth Riemannian manifold. For a suitable compact subset D⊂ M, we obtain the approximations to excursion probability ℙ\sup_{p∈ D} X(p) ≥ u \, as u→ ∞, for two cases: (i) X is smooth and isotropic; (ii) X is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands' approximation on Euclidean space which involves Pickands' constant and the volume of D. These extend the results in \citepCheng:2014 from sphere to general Riemannian manifolds.

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