The Annals of Probability

Climbing down Gaussian peaks

Robert J. Adler and Gennady Samorodnitsky

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


How likely is the high level of a continuous Gaussian random field on an Euclidean space to have a “hole” of a certain dimension and depth? Questions of this type are difficult, but in this paper we make progress on questions shedding new light in existence of holes. How likely is the field to be above a high level on one compact set (e.g., a sphere) and to be below a fraction of that level on some other compact set, for example, at the center of the corresponding ball? How likely is the field to be below that fraction of the level anywhere inside the ball? We work on the level of large deviations.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 1160-1189.

Received: January 2015
Revised: November 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60F10: Large deviations
Secondary: 60G60: Random fields 60G70: Extreme value theory; extremal processes 60G17: Sample path properties

Gaussian process excursion set large deviations exceedence probabilities topology


Adler, Robert J.; Samorodnitsky, Gennady. Climbing down Gaussian peaks. Ann. Probab. 45 (2017), no. 2, 1160--1189. doi:10.1214/15-AOP1083.

Export citation


  • Adler, R. J., Moldavskaya, E. and Samorodnitsky, G. (2014). On the existence of paths between points in high level excursion sets of Gaussian random fields. Ann. Probab. 42 1020–1053.
  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Anderson, E. J. (1983). A review of duality theory for linear programming over topological vector spaces. J. Math. Anal. Appl. 97 380–392.
  • Azaïs, J. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ.
  • Deuschel, J. and Stroock, D. W. (1989). Large Deviations. Pure and Applied Mathematics 137. Academic Press, Boston, MA.
  • Dunford, N. and Schwartz, J. T. (1988). Linear Operators. Part I: General Theory. Wiley, New York.
  • Luenberger, D. G. (1969). Optimization by Vector Space Methods. Wiley, New York.
  • Molchanov, I. and Zuyev, S. (2004). Optimisation in space of measures and optimal design. ESAIM Probab. Stat. 8 12–24.
  • Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • van der Vaart, A. W. and van Zanten, J. H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Inst. Math. Stat. Collect. 3 200–222. IMS, Beachwood, OH.