The Annals of Probability

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

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

Full-text: Open access


We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset\mathbb{R}^{d}$, with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets

\[A_{u}=\{t\in M:X;(t)>u\}\]

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 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.

Article information

Ann. Probab., Volume 41, Number 1 (2013), 134-169.

First available in Project Euclid: 23 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G52: Stable processes 60G60: Random fields
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G10: Stationary processes 60G17: Sample path properties

Infinitely divisible random fields moving average excursion sets extrema critical points Euler characteristic Morse theory geometry


Adler, Robert J.; Samorodnitsky, Gennady; Taylor, Jonathan E. High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Probab. 41 (2013), no. 1, 134--169. doi:10.1214/11-AOP738.

Export citation


  • [1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • [2] Adler, R. J., Taylor, J. E. and Worsley, K. J. (2013). Applications of Random Fields and Geometry: Foundations and Case Studies. Springer, New York. Early chapters available at
  • [3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [4] Braverman, M. and Samorodnitsky, G. (1998). Symmetric infinitely divisible processes with sample paths in Orlicz spaces and absolute continuity of infinitely divisible processes. Stochastic Process. Appl. 78 1–26.
  • [5] de Acosta, A. (1980). Exponential moments of vector-valued random series and triangular arrays. Ann. Probab. 8 381–389.
  • [6] Lancaster, P. (1969). Theory of Matrices. Academic Press, New York.
  • [7] Linde, W. (1986). Probability in Banach Spaces—Stable and Infinitely Divisible Distributions, 2nd ed. Wiley, Chichester.
  • [8] Maejima, M. and Rosiński, J. (2002). Type $G$ distributions on $\mathbb{R}^{d}$. J. Theoret. Probab. 15 323–341.
  • [9] Marcus, M. B. and Rosiński, J. (2005). Continuity and boundedness of infinitely divisible processes: A Poisson point process approach. J. Theoret. Probab. 18 109–160.
  • [10] Potter, H. S. A. (1940). The mean values of certain Dirichlet series, II. Proc. Lond. Math. Soc. (2) 47 1–19.
  • [11] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [12] Resnick, S. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.
  • [13] Rosiński, J. (1989). On path properties of certain infinitely divisible processes. Stochastic Process. Appl. 33 73–87.
  • [14] Rosiński, J. (1990). On series representations of infinitely divisible random vectors. Ann. Probab. 18 405–430.
  • [15] Rosiński, J. (1991). On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In Stable Processes and Related Topics (Ithaca, NY, 1990) (S. Cambanis, G. Samorodnitsky and M. S. Taqqu, eds.). Progress in Probability 25 27–41. Birkhäuser, Boston, MA.
  • [16] Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1014.
  • [17] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Chapman & Hall, New York.
  • [18] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.