Limit distributions for large P\'{o}lya urns
By Brigitte Chauvin, Nicolas Pouyanne, Reda Sahnoun. 2009
58 citations recorded by the source corpus.
We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings, the composition vector has asymptotically a first deterministic term of order $n$ and a second random term of order $n^{m/S}$. The object of interest is the limit distribution of this random term. The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree $m$. The limit laws appear to constitute a new family of probability densities supported by the whole real line.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/588fbbe182264dd53bd1f69ffbf2907ac97b10e7
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- A martingale approach for the elephant random walk (2017, methodology): “Baur and Bertoin [1] extensively used the connection to Pólya-type urns [12] as well as two functional limit theorems for multitype branching processes due to Janson [10], see also [4].”