Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

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