Mod-$\varphi$ convergence of Stirling distributions and limit theorems for zeros of their generating functions

TL;DR

This paper investigates the mod-$oldsymbol{ m }$ convergence of distributions involving Stirling numbers, deriving limit theorems for these distributions and the zeros of their generating functions, especially in the context of balls and boxes models.

Contribution

It introduces new limit theorems for Stirling-based distributions and analyzes the asymptotic distribution of zeros of their generating functions in a specific regime.

Findings

Identifies the asymptotic distribution of zeros for the generating polynomial of occupied boxes.

Establishes mod-$$ convergence for several probability distributions involving Stirling numbers.

Derives limit theorems for the distribution of zeros when the number of boxes grows linearly with the number of balls.

Abstract

We study mod-$\varphi$ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when $n$ balls are allocated equiprobably and independently among $\theta$ boxes in the regime when $\theta$ grows linearly with $n$.