  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 $F$ 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.
