Random Additions in Urns of Integers

Abstract

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, observing the values, then replacing the balls along with a new ball labeled with the sum of the two drawn balls. This model was introduced by Siegmund and Yakir (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.