Generalized gamma approximation with rates for urns, walks and trees

Peköz, Erol A ^[1] ; Röllin, Adrian ^[2] ; Ross, Nathan ^[3]
1. [1] Boston University
  
  Boston University
  
  City of Boston, Estados Unidos
2. [2] National University of Singapore
  
  National University of Singapore
  
  Singapur
3. [3] University of Melbourne
  
  University of Melbourne
  
  Australia
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 1776-1816
Idioma: inglés
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Resumen
- 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.