Limit shape of the generalized inverse gaussian-poisson distribution
File(s)
Preprint
OA Location
Author(s)
Bogachev, Leonid
Nuermaimaiti, Ruheyan
Voss, Jochen
Type
preprint
Abstract
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.
Date Issued
2023-03-14
Citation
arXiv, 2023
Journal / Book Title
arXiv
Copyright Statement
Copyright © 2023 The Author(s). This work is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).
License URL
Identifier
https://arxiv.org/abs/2303.08139
Subjects
count data
frequency distributions
sources/items models
generalized inverse Gaussian-Poisson distribution
Young diagrams
limit shape
informetrics data MSC 2020: Primary 62E17
Secondary 62P25