Statistical geometry of the smoothed random telegraph signal†

A study is made of various statistical properties of a particular non-Gaussian process, the low-pass filtered random telegraph signal. Results are obtained for the distribution of slope, the average numbers of zero and level crossings, the average number of crossings with a straight line, distributions of maxima and minima and the average number of specular points. As a prelude to these specific investigations, general expressions for the average numbers of crossings of random processes with arbitrary curves are derived. In particular, for the average number of zero-crossings of a stationary random process, the result does not explicitly involve the joint distribution of the process and its derivative.