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The Class of Tenable Zero-Balanced Pólya Urn Schemes: Characterization and Gaussian Phases

Part of: Combinatorial probability Combinatorics Stochastic processes Limit theorems

Published online by Cambridge University Press:  04 January 2016

Sanaa Kholfi  and
Hosam M. Mahmoud
Sanaa Kholfi*
Affiliation:
The George Washington University
Hosam M. Mahmoud*
Affiliation:
The George Washington University
*
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
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Abstract

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We study a class of tenable, irreducible, nondegenerate zero-balanced Pólya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal (left) eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of an urn starting with an asymptotically positive proportion of each color after jn draws. We identify three phases of jn: the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. We give interpretations for how the results in different phases conjoin at the ‘seam lines’. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory. We conclude with some illustrating examples.

Keywords

UrnPólya urncombinatorial probabilityphase transitionmultivariate martingale

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05A05: Permutations, words, matrices
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 702 - 728
Copyright
© Applied Probability Trust 

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