We discuss probability text models and its modifications. We have proved a theorem on the convergence of a multidimensional process of the number of urns containing a fixed number of balls in the Simon model to a multidimensional Gaussian process. We have introduced and investigated three two-parameter urn schemes that guarantee the power tail asymptotics of the number of occupied urns. These models do not follow the restriction on the limitation of the ratio of the number of urns with exactly one ball to the number of occupied urns that appears in the classical Bahadur-Karlin model.
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