In this paper we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical regularities in discrete data. Analytic properties of GNB distributions are studied. A GG distribution is proved to be a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GNB distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved 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. The class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Some applications of GNB distributions in meteorology are discussed.

中文翻译：

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作为混合几何定律和相关极限定理的广义负二项分布*

在本文中，我们研究了一个广泛而灵活的离散分布族，即所谓的广义负二项式 (GNB) 分布，它们是混合泊松分布，其中混合定律属于广义伽马 (GG) 分布类别。后者由 EW Stacy 引入，作为包含伽马、指数幂和威布尔分布的生命周期分布的特殊族。这些分布在许多真实现象的统计描述中似乎非常有前途，它们是描述离散数据中统计规律的非常方便且几乎通用的模型。研究了 GNB 分布的解析特性。当且仅当形状和指数幂参数不大于 1 时，GG 分布被证明是混合指数分布。混合分布被明确写出为集中在非负半线上的严格稳定定律的尺度混合。作为推论，GNB 分布的表示是作为混合几何分布获得的。考虑了具有随机成功概率的伯努利试验的相应方案。在该方案中，证明了泊松定理的随机模拟，建立了混合二项式分布向混合泊松定律的收敛性。证明了独立随机变量的随机和的极限定理，其中被加数的数量具有 GNB 分布并且被加数具有轻尾和重尾分布。极限定律的类别足够广泛，包括所谓的广义方差伽马分布。极限定律的各种表示是根据 Mittag-Leffler、Linnik 或 Laplace 分布的混合获得的。讨论了 GNB 分布在气象学中的一些应用。