Modern Stochastics: Theory and Applications

The structure of dependence between the forward and the backward recurrence times in a renewal process is considered. Monotonicity properties, as a function of time, for the tail of the bivariate distribution for the recurrence times are discussed, as well as their link with aging properties of the interarrival distribution F. A necessary and sufficient condition for the renewal function to be concave is also obtained. Finally, some properties of the conditional tail for one of the two recurrence times, given some information on the other, are studied. The results are illustrated by some numerical examples.

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.