Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,∞) such that for some T ≤ ∞, F*2(x, x+T] ∼ 2F(x, x+T]. The case T = ∞ corresponds to F being subexponential, and our analysis shows that the properties for T < ∞ are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.