We consider a two-color Pólya 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∕2m∕S. The object of interest is the limit distribution of this random term.
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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.
Publié le : 2011-02-15
Classification:
Pólya urn,
urn model,
martingale,
characteristic function,
embedding in continuous time,
multitype branching process,
Abelian integrals over Fermat curves,
60C05,
60J80,
05D40
@article{1292598027,
author = {Chauvin, Brigitte and Pouyanne, Nicolas and Sahnoun, Reda},
title = {Limit distributions for large P\'olya urns},
journal = {Ann. Appl. Probab.},
volume = {21},
number = {1},
year = {2011},
pages = { 1-32},
language = {en},
url = {http://dml.mathdoc.fr/item/1292598027}
}
Chauvin, Brigitte; Pouyanne, Nicolas; Sahnoun, Reda. Limit distributions for large Pólya urns. Ann. Appl. Probab., Tome 21 (2011) no. 1, pp. 1-32. http://gdmltest.u-ga.fr/item/1292598027/