IS THE SIBUYA DISTRIBUTION A PROGENY?

摘要

For 0 < a < 1, the Sibuya distribution sa is concentrated on the set N+ of positive integers and is defined by the generating function Sigma(infinity)(n=1) sa(n)z(n) = 1-(1-z)(a). A distribution q on N+ is called a progeny if there exists a branching process (Z(n))(n >= 0) such that Z(0) = 1, such that E(Z(1)) <= 1, and such that q is the distribution of Sigma(infinity)(n-0)Z(n). In this paper we prove that s(a) is a progeny if and only if 1/2 <= 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.