## The Annals of Probability

### Generalized gamma approximation with rates for urns, walks and trees

#### Abstract

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

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

Dates
Revised: February 2015
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410032

Digital Object Identifier
doi:10.1214/15-AOP1010

Mathematical Reviews number (MathSciNet)
MR3502594

Zentralblatt MATH identifier
06603562

#### Citation

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. https://projecteuclid.org/euclid.aop/1463410032

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