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Is the Sibuya distribution a progeny?

Part of: Series expansions Markov processes

Published online by Cambridge University Press:  12 July 2019

Gérard Letac
Gérard Letac*
Affiliation:
Université Paul Sabatier
*
*Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse, France. Email address: gerard.letac@math.univ-toulouse.fr
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Abstract

For 0 < a < 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a < 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

Keywords

Branching processprogenySibuya law

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 30B10: Power series (including lacunary series)
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 52 - 56
Copyright
© Applied Probability Trust 2019 

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References

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