Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

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