Random Telegraphic Signals with Fractal-like Probability Transition Rates

Many physical processes feature random telegraph signals, e.g., a time signal c(t) that randomly switches between two values over time. The present study focuses on the class of telegraphic processes for which the transition rates are formulated by using fractal-like expressions. By considering various restrictive hypotheses regarding the statistics of the waiting times, the present analysis provides the corresponding expressions of the unconditional and conditional probabilities, the mean waiting times, the mean phase duration, the autocorrelation function and the associated integral time scale, the spectral density, and the mean switching frequency. To assess the relevance of the various hypotheses, synthetically generated signals were constructed and used as references to evaluate the predictive quality of the theoretically derived expressions. The best predictions were obtained by considering that the waiting times probability density functions were Dirac peaks centered on the corresponding mean values.