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On the constant in the definition of subexponential distributions. (English. Russian original) Zbl 0971.60009
Theory Probab. Appl. 44, No. 2, 409-412 (1999); translation from Teor. Veroyatn. Primen. 44, No. 2, 455-458 (1999).
A distribution $$G$$ on $$[0, +\infty)$$ is said to be subexponential if it does not have a compact support, if for every $$y > 0$$, $$\lim_{x \to +\infty} G([x+y, +\infty))/G([x, +\infty)) = 1$$, and if $$\lim_{x \to +\infty} G*G([x, +\infty))/G([x, +\infty)) = c$$ for some constant $$c$$. Those distributions were introduced by V. P. Chistyakov [Theory Probab. Appl. 9, 640-648 (1964); translation from Teor. Veroyatn. Primen. 9, 710-718 (1964; Zbl 0203.19401)]. In this paper the author proves that the constant $$c$$ is necessarily equal to 2. The arguments use some Banach algebra methods, but mostly hinge upon some old and elementary results of W. Rudin [Ann. Probab. 1, 982-994 (1973; Zbl 0303.60014)]. It is to be noted that the proof is very different according to $$G$$ admitting a first moment or not.

##### MSC:
 60E05 Probability distributions: general theory 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
##### Keywords:
subexponential distribution
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