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

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.