Limit Shape of the Generalized Inverse Gaussian-Poisson Distribution

TL;DR

This paper characterizes the asymptotic limit shape and fluctuations of the generalized inverse Gaussian-Poisson distribution's diagrammatic representations, revealing normal fluctuations and Brownian motion approximations under specific growth conditions.

Contribution

It identifies the limit shape of scaled Young diagrams from GIGP samples and describes the asymptotic fluctuation behavior, extending understanding of this distribution's geometric properties.

Findings

Limit shape is an incomplete gamma function.

Fluctuations are asymptotically normal.

Empirical process approximates Brownian motion or Poisson process depending on growth regime.

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.