Smoothing Equations for Large Pólya Urns

Consider a balanced nontriangular two-color Pólya–Eggenberger urn process, assumed to be large, which means that the ratio $$\sigma $$σ of the replacement matrix eigenvalues satisfies $$1/2<\sigma <1$$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 $$W^\mathrm{DT}$$WDT in discrete time and $$W^\mathrm{CT}$$WCT in continuous time. The paper deals with the distributions of both $$W$$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 $$W^\mathrm{DT}$$WDT and $$W^\mathrm{CT}$$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 find upper bounds for the moments of $$W^\mathrm{DT}$$WDT and $$W^\mathrm{CT}$$WCT and we show that the laws of $$W^\mathrm{DT}$$WDT and $$W^\mathrm{CT}$$WCT are moment determined. We also prove that $$W^\mathrm{DT}$$WDT is supported by the whole real line, its exponential moment generating series has an infinite radius of convergence and $$W^\mathrm{DT}$$WDT admits a continuous density ($$W^\mathrm{CT}$$WCT was already known to have a density, infinitely differentiable on $$\mathbb R{\setminus }\{ 0\}$$R\{0} and not bounded at the origin).