Affine diminishing urns

We consider a large class of white-blue affine balanced urns diminished by the repeated drawing of multisets. We investigate the composition of the urn at different stages of drawing. Assuming the urn starts with $n$ balls, of which $\alpha n + g(n)$, with $\alpha \in [0,1]$ and $g(n) = o(n)$, are white, we find a major phase transition between a sublinear number of draws $j = o(n)$ and the linear case in which $j = \theta n + h(n)$. In both sublinear and linear phases, we get central limit theorems; however, the normalization in each phase is significantly different. The interplay of the different forces, such as $\theta$, $\alpha$, and the perturbation functions $g(n)$ and $h(n)$, enforce a number of restrictions and influences the parameterization in the central limit theorem. The methods of proof are based on recurrence, martingales, and asymptotic analysis. We then discuss two possible applications of this class. One application is a generalized OK Corral urn, and the other is on the dynamics of market depth in the stock market.