This paper investigates the defective renewal equations under the nonconvolution equivalent distribution class. The asymptotics of the solution to the defective renewal equations have been given for the heavy-tailed and light-tailed cases, respectively.
Kaiyong Wang. Yang Chen. Zhongquan Tan. "Asymptotics for the Solutions to Defective Renewal Equations." Abstr. Appl. Anal. 2014 (SI35) 1 - 5, 2014. https://doi.org/10.1155/2014/732735