Counting level crossings by a stochastic process

The number of times, NN, that a stochastic process has up-crossings of a fixed level within a fixed time interval, TT, is investigated. Existing integral formulas for the moments of NN 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 NN from the moment formulas. Particular attention is given to simplified results for the limiting situations of very small and very large values of TT, and to the behavior of variance, skewness, and kurtosis of NN. For small TT, the number NN approaches the well-known Poisson distribution, but the results for large TT are significantly different. For many stationary processes it is shown that the variance of NN tends to grow linearly with TT when TT is very large, but the large-TT growth rate is sometimes much smaller than that of the small-TT 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.