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Renewal process is a point process where an inter-event time betweensuccessive renewals is an independent and identically distributed randomvariable. Alternating renewal process is a dichotomous process and a slightgeneralization of the renewal process, where the inter-event time distributionalternates between two distributions. We investigate statistical properties ofthe number of renewals and occupation times for one of the two states inalternating renewal processes. When both means of the inter-event times arefinite, the alternating renewal process can reach an equilibrium. On the otherhand, an alternating renewal process shows aging when one of the meansdiverges. We provide analytical calculations for the moments of the number ofrenewals, occupation time statistics, and the correlation function for severalcase studies in the inter-event-time distributions. We show anomalousfluctuations for the number of renewals and occupation times when the secondmoment of inter-event time diverges. When the mean inter-event time diverges,distributional limit theorems for the number of events and occupation times areshown analytically. These are known as the Mittag-Leffler distribution and thegeneralized arcsine law in probability theory.

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