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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
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