A second-order asymptotic result for the probability of occurrence of a persistent and aperiodic recurrent event is given if the tail of the distribution of the waiting time for this event is regularly varying with index -1.

doi.org/10.2307/1427028, hdl.handle.net/1765/11459
Advances in Applied Probability
Erasmus School of Economics

Frenk, H. (1982). The Behavior of the Renewal Sequence in Case the Tail of the Waiting-Time Distribution Is Regularly Varying with Index -1. Advances in Applied Probability, 14(4), 870–884. doi:10.2307/1427028