## Electronic Communications in Probability

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

#### 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
Accepted: 25 January 2019
First available in Project Euclid: 22 March 2019

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

Digital Object Identifier
doi:10.1214/19-ECP214

#### 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

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