## The Annals of Probability

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

### Climbing down Gaussian peaks

Robert J. Adler and Gennady Samorodnitsky

#### Abstract

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

**Source**

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

**Dates**

Received: January 2015

Revised: November 2015

First available in Project Euclid: 31 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1490947316

**Digital Object Identifier**

doi:10.1214/15-AOP1083

**Mathematical Reviews number (MathSciNet)**

MR3630295

**Zentralblatt MATH identifier**

06797088

**Subjects**

Primary: 60G15: Gaussian processes 60F10: Large deviations

Secondary: 60G60: Random fields 60G70: Extreme value theory; extremal processes 60G17: Sample path properties

**Keywords**

Gaussian process excursion set large deviations exceedence probabilities topology

#### Citation

Adler, Robert J.; Samorodnitsky, Gennady. Climbing down Gaussian peaks. Ann. Probab. 45 (2017), no. 2, 1160--1189. doi:10.1214/15-AOP1083. https://projecteuclid.org/euclid.aop/1490947316