# Asymptotics of the overflow in urn models

## TL;DR

This paper investigates the asymptotic behavior of overflow counts in urn models with fixed capacities, extending previous work to general capacities and providing conditions for Poisson and normal limit distributions using probabilistic methods.

## Contribution

It generalizes prior results on overflow asymptotics from capacity one to arbitrary capacities, offering new probabilistic conditions for different limit distributions.

## Key findings

- Provides sufficient conditions for Poisson asymptotics.
- Provides sufficient conditions for normal asymptotics.
- Extends previous work from capacity one to general capacities.

## Abstract

Consider a number, finite or not, of urns each with fixed capacity $r$ and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain $r$ balls. When $r=1$, using analytic methods, Hwang and Janson gave conditions under which the overflow (which in this case is just the number of balls landing in non--empty urns) has an asymptotically Poisson distribution as the number of balls grows to infinity. Our aim here is to systematically study the asymptotics of the overflow in general situation, i.~e. for arbitrary $r$. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics for general $r$, thus extending Hwang--Janson's work. Our approach relies on purely probabilistic methods.