# Random Additions in Urns of Integers

## TL;DR

This paper introduces a novel urn model with integer labels, where balls are added based on sums of randomly drawn pairs, and proves an exponential limit law with a mean that depends on initial conditions.

## Contribution

It extends urn models to infinite integer labels, demonstrating an exponential limit law and employing the contraction method for recursive distributional equations.

## Key findings

- Convergence to an exponential distribution for the urn process.
- The exponential mean is a random variable influenced by initial configuration.
- The model combines multi-drawing with infinite types of balls.

## Abstract

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir ({\it Ann. Probab. 33(5), 2005}) for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.