Summary: Submitted to the Annals of Probability
HIGH LEVEL EXCURSION SET GEOMETRY FOR
NON-GAUSSIAN INFINITELY DIVISIBLE RANDOM
By Robert J. Adler Gennady Samorodnitsky,
and Jonathan E. Taylor,
Technion, Cornell and Stanford.
We consider smooth, infinitely divisible random fields X(t), t
M , M Rd
, with regularly varying L´evy measure, and are inter-
ested in the geometric characteristics of the excursion sets
Au = 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 vari-
ous types, of X in Au, conditional on Au being non-empty. 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 compli-