Central Limit Theorem for a Pólya-Friedman Mixed Urn Model
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918523839709184 |
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| author | Shi, Jianan Yin, Qing Miao, Yu |
| author_facet | Shi, Jianan Yin, Qing Miao, Yu |
| contents | This paper considers a two-color, single-draw urn model with two types of balls, denoted type $1$ and type $2$, with initial counts $Y^1_0\in N^+$ and $Y^2_0\in N^+$, respectively. At each discrete time step, a ball is drawn uniformly at random, its type observed, and then it is returned to the urn. The urn is subsequently updated according to a mixed replacement matrix: with fixed probability $p\in(0,1)$, the Friedman replacement matrix is applied, adding $a$ balls of the drawn type and $b$ balls of the opposite type; with fixed probability $1-p\in (0,1)$, the Pólya replacement matrix is applied, adding $c$ balls of the drawn type. We establish the central limit theorem for the proportion of type $1$ balls after $n$ draws. Furthermore, we provide corollaries that yield large deviation inequalities and the law of the iterated logarithm related to the proportion of type $1$ balls after $n$ draws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Central Limit Theorem for a Pólya-Friedman Mixed Urn Model Shi, Jianan Yin, Qing Miao, Yu Probability 60F05 This paper considers a two-color, single-draw urn model with two types of balls, denoted type $1$ and type $2$, with initial counts $Y^1_0\in N^+$ and $Y^2_0\in N^+$, respectively. At each discrete time step, a ball is drawn uniformly at random, its type observed, and then it is returned to the urn. The urn is subsequently updated according to a mixed replacement matrix: with fixed probability $p\in(0,1)$, the Friedman replacement matrix is applied, adding $a$ balls of the drawn type and $b$ balls of the opposite type; with fixed probability $1-p\in (0,1)$, the Pólya replacement matrix is applied, adding $c$ balls of the drawn type. We establish the central limit theorem for the proportion of type $1$ balls after $n$ draws. Furthermore, we provide corollaries that yield large deviation inequalities and the law of the iterated logarithm related to the proportion of type $1$ balls after $n$ draws. |
| title | Central Limit Theorem for a Pólya-Friedman Mixed Urn Model |
| topic | Probability 60F05 |
| url | https://arxiv.org/abs/2605.26669 |