Counting level crossings by a stochastic process

The number of times, N , that a stochastic process has up-crossings of a fixed level within a fixed time interval, T , is investigated. Existing integral formulas for the moments of N for a stationary Gaussian process are shown also to apply to processes that are neither stationary nor Gaussian, and explicit formulas are given for approximating the probability distribution of N from the moment formulas. Particular attention is given to simplified results for the limiting situations of very small and very large values of T , and to the behavior of variance, skewness, and kurtosis of N . For small T , the number N approaches the well-known Poisson distribution, but the results for large T are significantly different. For many stationary processes it is shown that the variance of N tends to grow linearly with T when T is very large, but the large- T growth rate is sometimes much smaller than that of the small- T Poisson process. More detailed results and some numerical examples are presented for the special case of a stationary Gaussian process crossing its own mean value.