On the finiteness of the moments of the measure of level sets of random fields

Diego Armentano (1) , Jean-Marc Azaïs (2) , Federico Dalmao (1) , Jose R. Léon (1) , Ernesto Mordecki (1)

1 UDELAR - Universidad de la República [Montevideo]
2 UT3 - Université Toulouse III - Paul Sabatier

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be 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, (iii) the boundedness of the joint density of the field 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. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

Mots clés: Rice formula; Moments of level sets; shot noise process; Moments of measure of level sets; Kac-Rice formula; Crofton formula

Domaines: Probabilités [math.PR]

Diego Armentano, Jean-Marc Azaïs, Federico Dalmao, Jose R. Léon, Ernesto Mordecki. On the finiteness of the moments of the measure of level sets of random fields. 2020. ⟨hal-02294101v2⟩