Theoretical Statistics and Mathematics Unit, ISI Delhi
We consider a class of urn models, where the selection probabilities are proportional to a weight function which is decreasing. We call these models negatively reinforced urns. We first considered the case where the weight function is linear. For such a class of models we show that the almost sure limit of the random configurations of the urns hold for any replacement matrix. We also establish almost sure limit of the colour counts. Further, central limit theorems are derived for the random configurations and also for the colour counts.
We also consider general decreasing weight functions. In particular, we show that for a doubly stochastic replacement matrix, uniform is the almost sure limit of the random configurations for weight functions satisfying certain general sufficient condition, covering a large class of examples including linear weight functions.