On rainflow cycles and the distribution of the number of interval crossings by a Markov chain

This paper treats the number of crossings of an interval by a Markov chain observed for finite time. The exact marginal distribution is derived, in the form of its probability generating function. A numerical example is presented. Interval crossings have an important application in fatigue of materials. For analysing complex fatigue loads, the so-called rainflow cycle counting method is widely used, which is equivalent to counting crossings of intervals.