High level excursion set geometry for non-Gaussian infinitely divisible random fields

Details High level excursion set geometry for non-Gaussian infinitely divisible random fields High level excursion set geometry for non-Gaussian infinitely divisible random fields

2009-07-20

arxiv.org/abs/0907.3359v1

Mathematics

Probability

43 pages

Scientific paper

We consider smooth, infinitely divisible random fields $X(t), t\in M)$, $M\subset \real^d$, with regularly varying L\'evy measure, and are interested in the geometric characteristics of the excursion sets \begin{eqnarray*} A_u = \{t\in M: X(t) >u\} \end{eqnarray*} over high levels $u$. For a large class of such random fields we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of $X$ in $A_u$, conditional on $A_u$ 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 complicated geometry. Whereas in the Gaussian case non-empty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.

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