The generalized inverse Gaussian-Poisson (GIGP) distribution proposed by Sichel in the 1970s has proved to be a flexible fitting tool for diverse frequency data, collectively described using the item production model. In this paper, we identify the limit shape (specified as an incomplete gamma function) of the properly scaled diagrammatic representations of random samples from the GIGP distribution (known as Young diagrams). We also show that fluctuations are asymptotically normal and, moreover, the corresponding empirical random process is approximated via a rescaled Brownian motion in inverted time, with the inhomogeneous time scale determined by the limit shape. Here, the limit is taken as the number of production sources is growing to infinity, coupled with an intrinsic parameter regime ensuring that the mean number of items per source is large. More precisely, for convergence to the limit shape to be valid, this combined growth should be fast enough. In the opposite regime referred to as "chaotic", the empirical random process is approximated by means of an inhomogeneous Poisson process in inverted time. These results are illustrated using both computer simulations and some classic data sets in informetrics.

翻译：Sichel于1970年代提出的广义逆高斯-泊松（GIGP）分布已被证明是适用于多种频率数据的灵活拟合工具，这些数据通常通过项目生产模型进行统一描述。本文确定了来自GIGP分布（称为杨图）的随机样本经适当缩放后的图示表示的极限形状（具体化为不完全伽马函数）。我们还证明了波动具有渐近正态性，并且对应的经验随机过程在逆时域中通过重新标度的布朗运动近似，其非均匀时间尺度由极限形状决定。这里，极限过程是在生产源数量趋于无穷大的同时，结合确保每个源的平均项目数足够大的内在参数机制下进行的。更精确地说，为了使收敛于极限形状成立，这种联合增长必须足够快速。在称为“混沌”的反向机制中，经验随机过程通过逆时域中的非齐次泊松过程近似。这些结果通过计算机模拟以及信息计量学中的一些经典数据集加以说明。