Abstract
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.
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Asmussen, S., Foss, S. & Korshunov, D. Asymptotics for Sums of Random Variables with Local Subexponential Behaviour. Journal of Theoretical Probability 16, 489–518 (2003). https://doi.org/10.1023/A:1023535030388
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DOI: https://doi.org/10.1023/A:1023535030388