Modern Stochastics: Theory and Applications

Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.