Gaussian fluctuations for the two urn model
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914907447885824 |
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| author | Kolesko, Konrad Sava-Huss, Ecaterina |
| author_facet | Kolesko, Konrad Sava-Huss, Ecaterina |
| contents | 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 $j\in\{1,\ldots,J\}$, $J\in\mathbb{N}$ added to the urns after $n$ draws. 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 $\sqrtρ$ and not at $ρ/2$, where $ρ$ is the dominant eigenvalue of the mean replacement matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08602 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gaussian fluctuations for the two urn model Kolesko, Konrad Sava-Huss, Ecaterina Probability Spectral Theory 60J80, 60J85, 60B20, 60F05, 60F17 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 $j\in\{1,\ldots,J\}$, $J\in\mathbb{N}$ added to the urns after $n$ draws. 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 $\sqrtρ$ and not at $ρ/2$, where $ρ$ is the dominant eigenvalue of the mean replacement matrix. |
| title | Gaussian fluctuations for the two urn model |
| topic | Probability Spectral Theory 60J80, 60J85, 60B20, 60F05, 60F17 |
| url | https://arxiv.org/abs/2301.08602 |