This article describes the Key renewal theorem (KRT) and Blackwell's renewal theorem (BRT). These two limit theorems appear in different forms, but can be shown to be equivalent. For ease of presentation, we follow the terminology used in Karlin and Taylor [1] and refer to them collectively as the renewal theorem (RT). The RT is a very useful tool for characterizing the asymptotic behavior of a probabilistic quantity of interest in a renewal process (RP). To start, we briefly review renewal

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... esses and the elementary renewal theorem (ERT). An RP is a counting process with independent and identically distributed (iid) inter-event times. Mathematically, it can be defined as follows. Let S 0=0 , and S n be the occurrence time of the nth event, n ≥ 1. Assume that 0 ≤ S 1 ≤ S 2 ≤ S 3 ≤ · · · Define X n = S n − S n−1 , n ≥ 1. Thus {X n , n ≥ 1} is a sequence of inter-event times. Next define Then N(t) is called an RP generated by {X n , n ≥ 1} if {X n , n ≥ 1} is a sequence of nonnegative iid random variables. We refer the readers to the article titled Definition and Examples of Renewal Processes in this encyclopedia for a detailed introduction and more examples of RPs. Suppose that the inter-event times {X n , n ≥ 1} have a common cumulative distribution function (cdf) G(·), that is, P(X n ≤ t) = G(t), n ≥ 1. Let τ and σ 2 represent the mean and variance of X 1 , respectively. To avoid triviality, we assume that G(0 − ) = 0 and G(0 + ) = G(0) < 1, which imply that τ > 0. Let