Asymptotic results for heavy-tailed distributions using defective renewal equations

In this note, we investigate the asymptotic behavior of two special functions that satisfy defective renewal equations and have a key role in risk theory. Heavy-tailed distributions are important in our analysis. Furthermore, applying our results to the renewal and classical models of risk theory yields a generalisation of a well-known asymptotic formula of [Embrechts, P., Veraverbeke, N., 1982. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72].