Limit Theorems for Renewal Processes

Abstract

Abstract This article describes the key renewal theorem and the Blackwell's renewal theorem. These two limit theorems for renewal processes are equivalent but of different forms. They are particularly useful for characterizing the asymptotic behavior of a probabilistic quantity of interest in a renewal process. We present two classical applications of these limit theorems: the limiting distributions of recurrence times in a renewal process and the asymptotic expansion of a renewal function. Further readings on the theoretical development of these limit theorems and their applications in different areas are also provided.