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

We consider smooth, infinitely divisible random fields (X (t), t is an element of M), M subset of R-d, with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets A(u) = {t is an element of M : X(t) > u} over high levels u. For a large class of such random fields, we compute the u -> infinity 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.