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Is the Sibuya distribution a progeny?

  • Gérard Letac (a1)

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.

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*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

References

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[1]Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
[2]Grey, D. R. (1975). Two necessary conditions for embeddability of a Galton–Watson branching process. Math. Proc. Cambridge Phil. Soc. 78, 339343.
[3]Harris, T. H. (1963). The Theory of Branching Processes. Springer, New York.
[4]Kozubowski, T. and Podgórski, K. (2018). A generalized Sibuya distribution. Ann. Inst. Stat. Math. 70, 855887.
[5]Sibuya, M. (1979). Generalized hypergeometric, digamma and trigamma distributions. Ann. Inst. Stat. Math. 31, 373390.
[6]Toulouse, P. S. (1999). Thèmes de Probabilités et Statistique. Dunod, Paris.
[7]Whittaker, E. T. and Watson, G. N. (1986). A Course in Modern Analysis. Cambridge University Press.

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