March 2023 On the finiteness of the moments of the measure of level sets of random fields
Diego Armentano, Jean Marc Azaïs, Federico Dalmao, José Rafael León, Ernesto Mordecki
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Braz. J. Probab. Stat. 37(1): 219-245 (March 2023). DOI: 10.1214/23-BJPS568


General conditions on smooth real valued random processes and fields are given to ensure the finiteness of the moments of the measure of their level sets. As a by product, a new generalized Kac–Rice formula for the expectation of the measure of these level sets in the one-dimensional case is obtained when the second moment is uniformly bounded. The conditions involve: (i) the differentiability of the trajectories up to a certain order k; (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order; and (iii) the boundedness of the field’s joint density and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. We also sketch the application of these tools to study critical points of random fields.


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Diego Armentano. Jean Marc Azaïs. Federico Dalmao. José Rafael León. Ernesto Mordecki. "On the finiteness of the moments of the measure of level sets of random fields." Braz. J. Probab. Stat. 37 (1) 219 - 245, March 2023.


Received: 1 December 2021; Accepted: 1 February 2023; Published: March 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580892
zbMATH: 1511.60074
Digital Object Identifier: 10.1214/23-BJPS568

Keywords: Crofton formula , Kac–Rice formula , Moments of measure of level sets , shot noise process

Rights: Copyright © 2023 Brazilian Statistical Association


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Vol.37 • No. 1 • March 2023
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