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Gaussian fluctuations for the two-urn model
- Part of
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- Journal:
- Advances in Applied Probability , First View
- Published online by Cambridge University Press:
- 02 December 2024, pp. 1-35
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We introduce a modification of the generalized Pólya urn model containing two urns, and we study the number of balls
$B_j(n)$ of a given color 
$j\in\{1,\ldots,J\}$ added to the urns after n draws, where 
$J\in\mathbb{N}$. We provide sufficient conditions under which the random variables 
$(B_j(n))_{n\in\mathbb{N}}$, properly normalized and centered, converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second-order behavior occurs at 
$\sqrt{\rho}$ and not at 
$\rho/2$, where 
$\rho$ is the dominant eigenvalue of the mean replacement matrix.
