Nonlinear randomized urn models

Works in preparation:

1. “Counting the number of equilibria in non-linear randomized urn models via phase transitions’’, with S.Laruelle (2021-)

Keywords: extended Pólya urn models, stochastic approximation, reinforcement, phase diagram, multiple equilibrium points.

In brief: Randomized urn models, originally devised for clinical trials, have been extensively investigated by various authors on ad hoc martingale arguments to solve a.s. convergence as well as its rate of convergence. This framework was revisited using Stochastic Approximation theory and extended for non uniform drawing rules. In the two-color setting using a convex skewing function \(f\) to reinforce the drawing rule, it is possible to have many asymptotic urn compositions regarding the components of the limit generating matrix and the behavior of \(f\). The purpose is to analyze these multiple targets depending on the model parameters to obtain explicit forms (if possible) and to better understand the phase transition. The results could be extended to higher dimensions.