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

TL;DR

This paper establishes conditions under which the moments of the measure of level sets of smooth random fields are finite, introduces a generalized Kac-Rice formula, and applies these results to various stochastic processes.

Contribution

It provides new general conditions ensuring finiteness of moments and derives a generalized Kac-Rice formula for level set measures of random fields.

Findings

Finiteness of moments for level set measures under specified conditions

A new generalized Kac-Rice formula for expected measure of level sets

Applications to shot noise, Gaussian, Chi-square, and diffusion processes

Abstract

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.