Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Abstract

Consider a generalized time-dependent Pólya urn process defined as follows. Let $$d\in \mathbb {N}$$ d ∈ N be the number of urns/colors. At each time n, we distribute $$\sigma _n$$ σ n 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 $$\mathcal {R}$$ R assuming some monotonicity and growth condition. The class $$\mathcal {R}$$ R includes convex functions and the classical case $$f(x)=x^{\alpha }$$ f ( x ) = x α , $$\alpha >1$$ 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.