## Asymptotics for sums of random variables with local subexponential behaviour.(English)Zbl 1033.60053

This paper studies distributions $$F$$ on $$[0,\,\infty)$$ such that for some $$T\leq\infty$$, $$F^{*2}(x,\,x+T]\sim 2F(x,\,x+T]$$. 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.

### MSC:

 60G50 Sums of independent random variables; random walks 60E05 Probability distributions: general theory
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