Introduction to Renewal Theory

Let {X i } i = 1 ∞ be a series of independent and identically distributed nonnegative random variables. Assume they are continuous. In particular, there exists some density function f X (x), x≥0, such that $$F_{X}(x) \equiv \mathrm{P}(X_{i} \leq x) =\int _{ t=0}^{x}f_{X}(t)\,dt$$ , i ≥ 1. Imagine X i representing the life span of a lightbulb. Specifically, there are infinitely many lightbulbs in stock. At time t = 0, the first among them is placed. It burns out after a (random) time of X 1. Then it is replaced by a fresh lightbulb that itself is replaced after an additional (random) time of X 2, etc. Note that whenever a new lightbulb is placed all statistically starts afresh. Let $$S_{n} = \Sigma _{i=1}^{n}X_{i}$$ , n ≥ 1, and set S 0 = 0. Of course, $$S_{n+1} = S_{n} + X_{n+1}$$ , n ≥0.