On the constant in the definition of subexponential distributions. (English. Russian original) Zbl 0971.60009

Rogozin, B. A.

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.

Reviewer: Thomas Simon (Berlin)

MSC:

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

Keywords:

subexponential distribution

Citations:

Zbl 0203.19401; Zbl 0303.60014