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) = x α , 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 any more. x=1 1 f (x) < ∞.