Consider a zero-delayed non-lattice renewal process. It is well known that "cycle-variables" associated with the renewal interval containing a particular time t (such as age at t or remaining life at t) tend in distribution to a limit as t goes to infinity, if the recurrence times have a finite mean. Also, it is known that the convergence is in (the much stronger) total variation, if in addition the n:th renewal time Sn has a density component for some n. 

First curiosity: This density condition is not needed for convergence in total variation of a certain cycle variable. 

Second curiosity: The finite mean condition is not necessary for the convergence in distribution of a certain cycle variable. 

Third curiosity: A stationary version of the process exists if the recurrence times have a finite mean. The simulation of that version (without any restriction on the process) seems to be impossible. But there is a simple simulation in the case when the recurrence times are bounded by a known constant.