Smoothing equations for large Pólya urns 1 May 29 th 2013

Consider a balanced non triangular two-color Pólya-Eggenberger 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 real-valued limit random variable arises, named W in discrete time and W 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 and W 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 and W and we show that the laws of W and W are moment-determined. We also prove that W is supported by the whole real line and admits a continuous density (W was already known to have a density, infinitely differentiable on R \ {0} and not bounded at the origin).