Central limit theorem analogues for multicolour urn models

Abstract

The asymptotic behaviour of a generalised P\'olya--Eggenberger urn is well--known to depend on the spectrum of its replacement matrix: If its dominant eigenvalue $r$ is simple and no other eigenvalue is `large' in the sense that its real part is greater than $r/2$, the normalized urn composition is asymptotically normally distributed. However, if there is more than one large eigenvalue, the first few random draws have a non--negligible effect on the evolution of the urn process and almost sure random tendencies of order larger than $\sqrt{n}$ typically prevent a classical central limit theorem. In the present work, a central limit theorem analogue for the fluctuations of urn models with regard to random linear drift and random periodic growth of order larger than $\sqrt{n}$ is proved, covering the $m$-ary search tree and B-trees. The proof builds on an eigenspace decomposition of the process in order to separate components of different growth orders. By an accurately tailored adaption of martingale techniques to the components, their joint limiting behaviour is established and translated back to the urn process. Conveniently, the approach encompasses results on small urn models and therefore provides a unifying perspective on central limit theorems for certain urn models, irrespective of their spectrum.