The Annals of Probability

Generalized gamma approximation with rates for urns, walks and trees

Erol A. Peköz, Adrian Röllin, and Nathan Ross

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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.

Article information

Ann. Probab. Volume 44, Number 3 (2016), 1776-1816.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 60E10: Characteristic functions; other transforms 60K99: None of the above, but in this section

Generalized gamma distribution Pólya urn model Stein’s method distributional transformations random walk random binary trees random plane trees preferential attachment random graphs


Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44 (2016), no. 3, 1776--1816. doi:10.1214/15-AOP1010.

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