Some properties of the crossings process generated by a stationary χ 2 process
Ken Sharpe · Advances in Applied Probability · 1978
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 → ∞.