GENERALIZED GAMMA APPROXIMATION WITH RATES FOR URNS, WALKS AND TREES

• 成果类型: Article
• 署名作者: Pekoz, Erol A.; Rollin, Adrian; Ross, Nathan
• 署名单位: Boston University; National University of Singapore; University of Melbourne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1010
• 发表日期: 2016
• 页码: 1776-1816
• 关键词: steins method; LIMIT-THEOREMS; models; size

摘要

We study a new class of time inhomogeneous Polya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein's method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.