  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.
