Some properties of the crossings process generated by a stationary χ2 process

Abstract

The process generated by the crossings of a fixed level, u, by the process P_n(t) is considered, where and the X_i(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of P_n(t) above u tends to a Rayleigh distribution as u→ ∞.