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 approximations to the excursion probabilities ℙ { sup p ∈ DX(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 Cheng and Xiao (Bernoulli 22, 1113–1130 2016) from spheres to general Riemannian manifolds.
Original language | English (US) |
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Pages (from-to) | 475-487 |
Number of pages | 13 |
Journal | Extremes |
Volume | 20 |
Issue number | 2 |
DOIs | |
State | Published - Jun 1 2017 |
Externally published | Yes |
Keywords
- Euler characteristic
- Excursion probability
- Gaussian fields
- Isotropic
- Locally isotropic
- Pickands’ constant
- Riemannian manifolds
ASJC Scopus subject areas
- Statistics and Probability
- Engineering (miscellaneous)
- Economics, Econometrics and Finance (miscellaneous)