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Pré-Publication, Document De Travail Année : 2013

Smoothing equations for large Polya urns

Brigitte Chauvin
Cécile Mailler

Résumé

Consider a balanced non triangular two-color P olya-Eggenberger urn process, assumed to be large which means that the ratio sigma of the replacement matrix eigenvalues satis es 1=2 < sigma < 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 real-valued limit random variable arises, named WDT in discrete time and WCT 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 WDT and WCT 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 nd upperbounds for the moments of WDT and WCT and we show that the laws of WDT and WCT are moment-determined. We also prove that their densities are not bounded at the origin.
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Dates et versions

hal-00785722 , version 1 (06-02-2013)

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  • HAL Id : hal-00785722 , version 1

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Brigitte Chauvin, Cécile Mailler, Nicolas Pouyanne. Smoothing equations for large Polya urns. 2013. ⟨hal-00785722⟩
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