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Pólya Urns Via the Contraction Method

Published online by Cambridge University Press:  01 September 2014

MARGARETE KNAPE  and
RALPH NEININGER
MARGARETE KNAPE
Affiliation:
Institute for Mathematics, J. W. Goethe University, 60054 Frankfurt a.M., Germany (e-mail: knape@math.uni-frankfurt.de, neiningr@math.uni-frankfurt.de)
RALPH NEININGER
Affiliation:
Institute for Mathematics, J. W. Goethe University, 60054 Frankfurt a.M., Germany (e-mail: knape@math.uni-frankfurt.de, neiningr@math.uni-frankfurt.de)
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Abstract

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.

Keywords

Primary 60C05Secondary 60F0560J0568Q25

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 1148 - 1186
Copyright
Copyright © Cambridge University Press 2014 

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