Distributions at random events

Abstract

We review the generalized Rice formula approach
to deriving long-run distributions of a variety of characteristics defined at random events defined on a stochastic process. While the approach stems from the same principle originally introduced by Rice for the level crossing intensity in a random signal, we show how it generalizes to more general contexts. Firstly, we discuss events defined on random surfaces through crossing levels of  possibly multivariate valued stochastic fields. Secondly, the dynamics is introduced by adding time
argument and introducing the concept of velocity measured at moving surface. Thirdly, extensions  beyond the Gaussian model are shown by presenting effective models for sampling from the
distribution of a non-Gaussian noise observed at instances of level crossing by a process driven by this noise. The importance of these generalization for engineering applications is illustrated through examples.