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Pólya urns via the contraction method. (English) Zbl 1301.60012
Summary: We propose an approach to analysing the asymptotic behaviour of Pólya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.

MSC:
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
68Q25 Analysis of algorithms and problem complexity
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