Introduction to Renewal Theory

Abstract

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.