Polya Urn Models
John Haigh · Journal of the Royal Statistical Society Series A (Statistics in Society) · 2009
Since the classical treatise by Johnson and Kotz (1977), many new results and applications of urn models have been developed. This timely monograph presents an overview of the current state of this field. It is not a book for beginners, even though the first two chapters develop much of discrete probability ab initio via urn models. Familiarity with the content of (at least) a Bachelor’s degree that includes standard probability distributions, and the different ideas of convergence of random variables, is assumed. All 10 chapters end with a set of exercises, with full solutions over 43 pages. The main focus is on models where balls take one of k different colours. At each drawing, one ball is selected completely at random from the urn: suppose that its colour is i. This ball is then returned to the urn, along with Aij balls of colour j (j = 1,2,…,k). The matrix A=(Aij) is termed a schema; its entries are necessarily integers, but they may take negative values. Interest is on the development of the urn’s composition, especially asymptotic results, so there are strong restrictions on the form of A, to ensure that the process can continue indefinitely, whatever balls are drawn—the model must be tenable.