Electronic Communications in Probability

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

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

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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

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

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

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

Gaussian fields nodal sets Crofton formula Kac-Rice formula

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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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