Negatively reinforced balanced urn schemes
Advances in Applied Mathematics
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 random replacement rule and the weight function w, such as, w is differentiable and w(0)<∞, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if either the number of colours are sufficiently large or if the limiting replacement matrix is a doubly stochastic matrix with diagonal entries more than 1/2, the random configuration of the urn converges to the uniform vector almost surely and the asymptotic normality holds.
Kaur, Gursharn, "Negatively reinforced balanced urn schemes" (2019). Journal Articles. 898.