The class of tenable zero-balanced Pólya urn schemes: characterization and Gaussian phases

We study a class of tenable irreducible nondegenerate zero-balanced Pólya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal left eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of the urn after j_n draws. We identify three phases of j_n: The growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. These Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory.