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Gaussian Phases Toward Statistical Equilibrium in Some Urn Models Open Access

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We are interested in a group of urn models with mean reverting property, that is, they reach statistical equilibrium, usually at the mean, when scale space and/or time reach infinity. Both Ehrenfest and Bernoulli-Laplace urns are within this category. We analyze these urns in different phases (sublinear, linear and superlinear), along which the models approach equilibrium, and try to identify and explain the cutoff phenomenon (Diaconis, 1996) and the changes at the "seamlines" between phases (Balaji, Mahmoud and Zhang, 2010). Essentially, we want to investigate the probabilitylaws of all the embedded phases. Analyzing the phases will help us further understand the nature of these models.We introduce a generalization of both the Ehrenfest and Bernoulli-Laplace urns, in which samples of independent identically distributed random sizes with a general generating discrete distribution (the generator) are taken out of an urn (Ehrenfest) or urns (Bernoulli-Laplace) at each time epoch. Generators with the same mean and variance exert the same influence on the phases. The influence of the generators gets attenuated from the sublinear phase to the linear phase and disappears at the superlinear phase.This methodology can be generalized to study other similar random structures and is left for future developments.

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