Open Access
2019 Conditions for the finiteness of the moments of the volume of level sets
D. Armentano, J-M. Azaïs, D. Ginsbourger, J.R. León
Electron. Commun. Probab. 24: 1-8 (2019). DOI: 10.1214/19-ECP214


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.


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D. Armentano. J-M. Azaïs. D. Ginsbourger. J.R. León. "Conditions for the finiteness of the moments of the volume of level sets." Electron. Commun. Probab. 24 1 - 8, 2019.


Received: 1 November 2018; Accepted: 25 January 2019; Published: 2019
First available in Project Euclid: 22 March 2019

zbMATH: 1412.60057
MathSciNet: MR3933041
Digital Object Identifier: 10.1214/19-ECP214

Primary: 20B15 , 60D05 , 60G15 , 60G60

Keywords: Crofton formula , Gaussian fields , Kac-Rice formula , nodal sets

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