Statistics of the number of renewals, occupation times and correlation in ordinary, equilibrium and aging alternating renewal processes

Renewal process is a point process where an inter-event time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the inter-event time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the inter-event times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the inter-event-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of inter-event time diverges. When the mean inter-event time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.