Smoothing Equations for Large Plya Urns

Consider a balanced nontriangular two-color Plya-Eggenberger urn process, assumed to be large, which means that the ratio of the replacement matrix eigenvalues satisfies . 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 in discrete time and in continuous time. The paper deals with the distributions of both . 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 and 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 upper bounds for the moments of and and we show that the laws of and are moment determined. We also prove that is supported by the whole real line, its exponential moment generating series has an infinite radius of convergence and admits a continuous density ( was already known to have a density, infinitely differentiable on and not bounded at the origin).