## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 24 (2019), paper no. 17, 8 pp.

### Conditions for the finiteness of the moments of the volume of level sets

D. Armentano, J-M. Azaïs, D. Ginsbourger, and J.R. León

#### Abstract

Let $X(t)$ be a Gaussian random field $\mathbb R^d\to \mathbb R$. Using the notion of $(d-1)$-*integral geometric measures*, we establish a relation between (a) the volume of level sets, and (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.

#### Article information

**Source**

Electron. Commun. Probab., Volume 24 (2019), paper no. 17, 8 pp.

**Dates**

Received: 1 November 2018

Accepted: 25 January 2019

First available in Project Euclid: 22 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1553220033

**Digital Object Identifier**

doi:10.1214/19-ECP214

**Subjects**

Primary: 60G15: Gaussian processes 60G60: Random fields 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 20B15: Primitive groups

**Keywords**

Gaussian fields nodal sets Crofton formula Kac-Rice formula

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Armentano, D.; Azaïs, J-M.; Ginsbourger, D.; León, J.R. Conditions for the finiteness of the moments of the volume of level sets. Electron. Commun. Probab. 24 (2019), paper no. 17, 8 pp. doi:10.1214/19-ECP214. https://projecteuclid.org/euclid.ecp/1553220033