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Smoothing equations for large Pólya urns
, 2013
"... Consider a balanced non triangular twocolor PólyaEggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried by ..."
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Consider a balanced non triangular twocolor PólyaEggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector admits also an almost sure asymptotics and a realvalued limit random variable arises, named W DT in discrete time and W CT in continous time. The paper deals with the distributions of both W. Appearing as martingale limits, known to be nonnormal, these laws remain up to now rather mysterious. Exploiting the underlying tree structure of the urn process, we show that W DT and W CT are the unique solutions of two distributional systems in some suitable spaces of integrable probability measures. These systems are natural extensions of distributional equations that already appeared in famous algorithmical problems like Quicksort analysis. Existence and unicity of the solutions of the systems are obtained by means of contracting smoothing transforms. Via the equation systems, we find upperbounds for the moments of W DT and W CT and we show that the laws of W DT and W CT are momentdetermined. We also prove that their densities are not bounded at the origin.
Pólya urns via the contraction method
, 2013
"... We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distr ..."
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We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. 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 nonnormal limit distributions and with asymptotic periodic distributional behavior.
Describing the asymptotic behaviour of multicolour Pólya urns via smoothing systems analysis
, 2014
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14 Urn models and applications
"... Description: An urn contains a number of balls of different types (colours), changing successively over time, typically drawing a ball from the urn and returning it together with some further balls of colours depending on the colour drawn. The simplest case of two colours and each time adding one ex ..."
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Description: An urn contains a number of balls of different types (colours), changing successively over time, typically drawing a ball from the urn and returning it together with some further balls of colours depending on the colour drawn. The simplest case of two colours and each time adding one extra ball of the colour drawn is the classical Pólya urn model, a nice application of martingale methods, which is treated in textbooks such as Feller. There is a wide literature on further developments, more recent contributions by Janson and by Chauvin et al. give overviews that can lead the direction for this project. Applications include Dirichlet processes with connections to Bayesian statistics and reinforced random walks and other interacting urn models of relevance in Biology and Economics. References:
9. Bibliography............................................................................111. Team Research Scientist
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1