In applied probability one is often interested in the asymptotic behavior of a certain quantity. If a regenerative phenomenon can be imbedded, then one has the problem that the event of interest may have occurred but cannot be observed at the renewal points. In this paper an extension to the renewal theorem is proved which shows that the quantity of interest converges. As an illustration an open problem in risk theory is solved.
"An extension to the renewal theorem and an application to risk theory." Ann. Appl. Probab. 7 (1) 121 - 133, February 1997. https://doi.org/10.1214/aoap/1034625255