Generalized gamma approximation with rates for urns, walks and trees. (English) Zbl 1367.60019

A generalized gamma distribution \(\mathrm{GG}(\alpha,\beta)\) for \(\alpha,\beta>0\) arises as the distribution of \(X^{1/\beta}\) when the random variable \(X\) is gamma distributed with parameter \(\alpha\). The following Pólya type urn model is investigated. Let \(P^\ell_n(1,w)\) denote the distribution of the number of white balls in an urn after \(n\) draws when starting with one black and \(w\) white balls, and when after each draw the ball is replaced by two balls of the same color and after every \(\ell\)-th draw an additional black ball is added. The main result of the paper shows that the Kolmogorov distance of the distribution of \(N_n/\mu_n\) to \(\mathrm{GG}(w,\ell+1)\), where \(N_n\) is \(P^\ell_n(1,w)\)-distributed, can be bounded from above and below by a constant times \(n^{-\ell/(\ell+1)}\). The normalizing constants \(\mu_n\) depend on \(\mathbb E[N_n^{\ell+1}]\) and are explicitly given. The main technique of proof is an application of Stein’s method for log-concave densities (to which the generalized gamma densities belong) together with a characterization of generalized gamma distributions as fixed points of distributional transformations related to generalized equilibrium distributions. The result is a significant generalization to numerous previous works, e.g., for \(\ell=1\) it gives the rate of convergence in Example 3.1 of S. Janson [Probab. Theory Relat. Fields 134, No. 3, 417–452 (2006; Zbl 1112.60012)]. It can be directly applied to certain preferential attachment random graph models. Some of the urn models can be embedded into certain models for the size of random subtrees and into certain local times of random walks and random walk bridges as well as excursions and meanders of the latter. This also enables the authors to prove the convergence to generalized gamma distributions with optimal rates in these settings.


60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
60E10 Characteristic functions; other transforms
60K99 Special processes


Zbl 1112.60012
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