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Limit distributions for large Pólya urns. (English) Zbl 1220.60006
The authors consider a two-color Pólya urn in the case when a fixed number \(S\) of balls is added at each step. The urn starts with a finite number of red and black balls, and at every step \(n\) one draws a ball uniformly at random, notices its color, and puts the ball back into the urn. In addition, if the ball is red, then one adds \(a\) red and \(b\) black balls, and if the ball is black, one adds \(c\) red and \(d\) black balls into the urn. Thus, the model is encoded by a matrix \(R={a\;b\choose c\;d}\). The model is balanced in the sense that \(a+b=c+d=S\).
Assume that the urn is large, that is, the second eigenvalue \(m\) of the replacement matrix \(R\) satisfies \(1/2<m /S\leq 1\). After \(n\) drawings, the composition vector (i.e., the vector whose components are ball numbers at time \(n\)) has asymptotically a first deterministic term of order \(n\) and a second random term of order \(n^{m/S}\). The object of interest of the paper 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.

MSC:
60C05 Combinatorial probability
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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