Theoretical Statistics and Mathematics Unit, ISI Delhi
We consider a negatively reinforced urn model, where a ball is selected at each time according to a decreasing weight function. We show that there exists a coupling between negatively reinforced urn model and the classical urn model, when the weight function is linear. We also obtain the almost sure convergence of the random urn configuration and the color count statistics of a negatively reinforced urn model for both linear and non-linear weight functions. In particular, we get almost sure convergence to a degenerate uniform vector whenever the replacement matrix is doubly stochastic. Under certain assumptions, we obtain the central limit theorem type results for a general weight function using the method of stochastic approximation. Further, to investigate the two choice paradigm, we introduce a model of two choices with negative reinforcement, which in-effect leads to a co-operative dynamical system. For this model, we show that the almost sure limit of the random urn configuration is also uniform but with a significantly improved limiting variability.
Parts are joint work with Antar Bandyopadhyay.