Smoothing equations for large P\'olya urns

TL;DR

This paper investigates the distributional properties of limits in large two-color Pólya urns, deriving unique solutions to associated distributional equations, and establishing moment bounds, support, and density characteristics.

Contribution

It introduces a novel approach using smoothing transforms to characterize the distributions of urn limit variables WDT and WCT, proving their uniqueness and moment-determinacy.

Findings

WDT and WCT are unique solutions to distributional systems.

WDT has support on the entire real line and a continuous density.

Moment bounds and moment-determinacy of WDT and WCT are established.

Abstract

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 satisfies 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 find 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 WDT is supported by the whole real line and admits a continuous density (WCT was already known to have a density, infinitely differentiable on R\{0} and not bounded at the origin).