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.

Published at http://dx.doi.org/10.1214/15-AOP1010 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• A survey of random processes with reinforcement
• Analytic urns
• Asymptotic theorems of sequential estimation-adjusted urn models
• Degree asymptotics with rates for preferential attachment random graphs
• Local limit theorems via Landau-Kolmogorov inequalities
• Total variation error bounds for geometric approximation
• Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model
• Zero biasing and a discrete central limit theorem

Cited by in corpus (3)

• Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions
• Concentration inequalities from monotone couplings for graphs, walks, trees and branching processes
• Rate of convergence for traditional Pólya urns