On a class of zero-balanced urn models Open Access
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We study a class of tenable irreducible nondegenerate zero-balanced polya 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≥1 balls from the initial condition to the steady state. We look at the status of an urn starting with a positive proportion of each color after jn draws. We identify three phases of jn: The growing sublinear, the linear, and the superlinear. In the absence of an initially dominant subset of colors, the number of balls of all colors in the growing sublinear phase 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. We give interpretations for how the results in different phases conjoin at the ``seamlines." In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory.The case of an initially dominant subset of colors poses challenges requiring finer asymptotic analysis in the sublinear phase. We characterize noncritical cases with an initially dominant subset of colors in which not all ball counts satisfy one multivariate central limit theorem, but rather a subset of the ball counts satisfies a singular multivariate central limit theorem. The rest of the cases are critical, in which all the ball counts satisfy a multivariate central limit theorem, but under a different scaling. However, for these critical cases the Gaussian phases are delayed considerably. We conclude with some illustrating examples and further developments.