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

@article{Pekoz2016GeneralizedGA, title={Generalized gamma approximation with rates for urns, walks and trees}, author={Erol A. Pekoz and Adrian R{\"o}llin and Nathan Ross}, journal={Annals of Probability}, year={2016}, volume={44}, pages={1776-1816} }

We study a new class of time inhomogeneous Polya-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…

## 32 Citations

Pólya urns with immigration at random times

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We study the number of white balls in a classical P\'olya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times…

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It is proved that the law of the south-east corner of a triangular Young tableau follows asymptotically a product of generalized gamma distributions, which allows us to tackle some questions related to the continuous limit of large random Young tableaux and links with random surfaces.

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Generalized Mittag Leffler distributions arising as limits in preferential attachment models

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For 0 �, let (S � �,�+r ){r�0} denote an in- creasing(decreasing) sequence of variables forming a time inhomogeneous Markov chain whose marginal distributions are equivalent to generalized Mittag…

Periodic Pólya Urns and an Application to Young Tableaux

- MathematicsAofA
- 2018

It is proved that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions, which can also be seen as powers of Gamma distributions.

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The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized…

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Abstract We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak…

## References

SHOWING 1-10 OF 80 REFERENCES

Stein's method for the half-normal distribution with applications to limit theorems related to simple random walk

- Mathematics
- 2013

We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of symmetric, simple random walk: the…

Stein's method for the half-normal distribution with applications to limit theorems related to the simple symmetric random walk

- Mathematics
- 2013

We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of the simple symmetric random walk: the…

Degree asymptotics with rates for preferential attachment random graphs

- Mathematics
- 2013

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show…

Total variation error bounds for geometric approximation

- Mathematics
- 2010

We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework…

Limit theorems for triangular urn schemes

- Mathematics
- 2006

Abstract.We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic…

New rates for exponential approximation and the theorems of Rényi and Yaglom

- MathematicsThe Annals of Probability
- 2011

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of…

Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

- Mathematics
- 2011

We give a unified treatment of the limit, as the size tends to infinity,
of simply generated random trees,
including both
the well-known result in the standard case
of critical Galton–Watson…

Joint degree distributions of preferential attachment random graphs

- MathematicsAdvances in Applied Probability
- 2017

Abstract We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak…

Analytic urns

- Mathematics
- 2003

This article describes a purely analytic approach to urn models of the generalized or extended Pólya–Eggenberger type, in the case of two types of balls and constant “balance,” that is, constant row…

Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model

- Mathematics
- 2012

We propose a way of finding a Stein type characterization of a given absolutely continuous distribution $\mu$ on $\R$ which is motivated by a regression property satisfied by an exchangeable pair…