Journal Article

HIGH LEVEL EXCURSION SET GEOMETRY FOR NON-GAUSSIAN INFINITELY DIVISIBLE RANDOM FIELDS

Samorodnitsky, Gennady, Taylor, Jonathan E., Adler, Robert J.

2013

Overview

We consider smooth, infinitely divisible random fields (X(t), t ∈ M), M ⊂ ℝ d , with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets A u = {t ∈ M : X(t) > u} over high levels u. For a large class of such random fields, we compute the u → ∞ asymptotic joint distribution of the numbers of critical points, of various types, of X in A u , conditional on A u being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 60D05 / 60G10 / 60G17 / 60G52 / 60G60 / Asymptotic methods / Critical points / Cubes / Eigenvalues / Euler characteristic / Eulers equations / excursion sets / extrema / Geometry / Infinitely divisible random fields / Mathematical functions / Mathematical theorems / Morse theory / moving average / Normal distribution / Partial derivatives / Probability distribution / Random variables / Second order partial derivatives / Studies / Sufficient conditions