# Random Additions in Urns of Integers

Consider an urn containing balls labeled with integer values. Define a process by drawing two balls, observing the values, then replacing the balls in the urn along with a new ball labeled with the sum of the two drawn balls. What does the configuration of the urn look like after many rounds? The surprising result is an exponential limit law for the empirical distribution defined by the urn. The mean is a random variable which depends on the starting configuration. I will outline the proof of convergence, which uses the contraction method for recursive distributional equations. I will also discuss some other interesting urn models that include the features of multi-drawing and an infinite type of balls.