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On generalized Pólya urn models. (English) Zbl 1290.60009
Summary: We study an urn model introduced in the paper of M. Chen and C. Wei [J. Appl. Probab. 42, No. 4, 964–976 (2005; Zbl 1093.60007)], where at each discrete time step $$m$$ balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case $$m = 1$$. We provide exact expressions for the expectation and the variance of the number of white balls after $$n$$ draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the $$m$$ balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman’s urn model.

##### MSC:
 60C05 Combinatorial probability 60F05 Central limit and other weak theorems 05A15 Exact enumeration problems, generating functions
##### Keywords:
urn model; limiting distribution
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##### References:
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