Negatively Reinforced Balanced Urn Schemes.

We consider weighted negatively reinforced urn schemes with finitely many colours. An urn scheme is called negatively reinforced, if the selection probability for a colour is proportional to the weight $w$ of the colour proportion, where $w$ is a non-increasing function. Under certain assumptions on the replacement matrix $R$ and weight function $w$, such as, $w$ is differentiable and $w(0) < \infty$, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if $R$ is doubly stochastic the random configuration of the urn converges to the uniform vector, and asymptotic normality holds, if the number of colours in the urn are sufficiently large.