The authors study analytic properties of the so called generalized negative binomial (GNB) distribution \[ P(N_{r,\alpha,\mu}=k)=\int_0^\infty \frac{z^k}{k!}e^{-z}g^*(z;r,\alpha,\mu)dz, \quad k=0,1,2,\dots,\ r>0,\ \alpha\in \mathbb{R},\ \mu>0, \] where \[ g^*(x;r,\alpha,\lambda)=\frac{|\alpha| \lambda^r}{\Gamma(r)}x^{\alpha r-1}e^{-\lambda x^\alpha},\quad x\geq0,\ \alpha\in \mathbb{R},\ \lambda>0,\ r>0, \] is the density of the generalized gamma (GG) distribution, first described as a unitary family by E. Stacy [Ann. Math. Stat. 33, 1187–1192 (1962; Zbl 0121.36802)]. The authors show that the GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. They write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, a representation follows of the GNB distribution as a mixed geometric distribution. They also consider the corresponding scheme of Bernoulli trials with random probability of success, and they prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws.
The authors prove limit theorems for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light and heavy-tailed distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. It is stated explicitly in the paper that the GG distribution contains almost all the absolutely continuous distributions concentrated on the non-negative half-line, and, hence, the GNB distribution is a very flexible and potentially most applicable family of discrete distributions. Applications of the GNB distribution in meteorology are discussed.