Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Abstract

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N be the number of urns/colors. At each time n, we distribute σn balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R assuming some monotonicity and growth condition. The class R includes convex functions and the classical case f(x)=xα, α>1. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.