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We study distributions $F$ on $[0,\infty)$ such that for some $T\le\infty$,$F^{*2}(x,x+T]\sim 2 F(x,x+T]$. The case $T=\infty$ corresponds to $F$ beingsubexponential, and our analysis shows that the properties for $T<\infty$ are,in fact, very similar to this classical case. A parallel theory is developed inthe presence of densities. Applications are given to random walks, the keyrenewal theorem, compound Poisson process and Bellman-Harris branchingprocesses.

Asymptotics for sums of random variables with local subexponential behaviour