Pólya Urn Models under General Replacement Schemes

In this paper, we consider a Pólya urn model containing balls of m different labels under a general replacement scheme, which is characterized by an m×m addition matrix of integers without constraints on the values of these m integers other than non-negativity. This urn model includes some important urn models treated before. By a method based on the probability generating functions, we consider the exact joint distribution of the numbers of balls with particular labels which are drawn within n draws. As a special case, for m = 2, the univariate distribution, the probability generating function and the expected value are derived exactly. We present methods for obtaining the probability generating functions and the expected values for all n exactly, which are very simple and suitable for computation by computer algebra systems. The results presented here develop a general workable framework for the study of Pólya urn models and attract our attention to the importance of the exact analysis. Our attempts are very useful for understanding non-classical urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results.