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Is the Sibuya distribution a progeny?
- Part of
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- Journal:
- Journal of Applied Probability / Volume 56 / Issue 1 / March 2019
- Published online by Cambridge University Press:
- 12 July 2019, pp. 52-56
- Print publication:
- March 2019
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- Article
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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.
