IEOR 6711 : Introduction to Renewal Theory II

Here we will present some deeper results in renewal theory such as a central limit theorem for counting processes, stationary versions of renewal processes, renewal equations, the key renewal theorem, weak convergence. 1 Central limit theorem for counting processes Consider a renewal process {t n : n ≥ 1} with iid interarrival times X n = t n − t n−1 , n ≥ 0, such that 0 < E(X) = 1/λ < ∞ and 0 < σ 2 = V ar(X) < ∞. Let N (t) = max{n : t n ≤ t}, t ≥ 0, denote the counting process. =⇒ denotes convergence in distribution (weak convergence). Z denotes a rv with a N (0, 1) distribution (standard unit normal);