Asymptotics for sums of random variables with local subexponential behaviour

Abstract

We study distributions $F$ on $[0,\infty )$ such that for some $T\le \infty$, $F^{\ast 2}(x,x+T]\sim 2F(x,x+T]$. The case $T=\infty$ corresponds to $F$ being subexponential, and our analysis shows that the properties for $T<\infty$ 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.