Analytic urns

• 成果类型: Article
• 署名作者: Flajolet, P; Gabarró, J; Pekari, H
• 署名单位: Universitat Politecnica de Catalunya
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000026
• 发表日期: 2005
• 页码: 1200-1233
• 关键词: CENTRAL LIMIT-THEOREMS; generalized polya urn; 2-3 trees; models

摘要

This article describes a purely analytic approach to urn models of the generalized or extended Polya-Eggenberger type, in the case of two types of balls and constant balance, that is, constant row sum. The treatment starts from a quasilinear first-order partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences in the case of subtractive urns are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limit and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x(h) + y(h) = 1. Several urn models, including a classical one associated with balanced trees (2-3 trees and fringe-balanced search trees) and related to a previous study of Panholzer and Prodinger, as well as all urns of balance I or 2 and a sporadic urn of balance 3, are shown to admit of explicit representations in terms of Weierstra beta elliptic functions: these elliptic models appear precisely to correspond to regular tessellations of the Euclidean plane.