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May 2016 Generalized gamma approximation with rates for urns, walks and trees
Erol A. Peköz, Adrian Röllin, Nathan Ross
Ann. Probab. 44(3): 1776-1816 (May 2016). DOI: 10.1214/15-AOP1010


We study a new class of time inhomogeneous Pólya-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.


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Erol A. Peköz. Adrian Röllin. Nathan Ross. "Generalized gamma approximation with rates for urns, walks and trees." Ann. Probab. 44 (3) 1776 - 1816, May 2016.


Received: 1 September 2013; Revised: 1 February 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1367.60019
MathSciNet: MR3502594
Digital Object Identifier: 10.1214/15-AOP1010

Primary: 60C05 , 60F05
Secondary: 60E10 , 60K99

Keywords: distributional transformations , generalized gamma distribution , Pólya urn model , preferential attachment random graphs , random binary trees , random plane trees , Random walk , Stein’s method

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • May 2016
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