Friedman vs Pólya
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917181506191360 |
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| author | Alves, Raphael Rosales, Rafael A. |
| author_facet | Alves, Raphael Rosales, Rafael A. |
| contents | Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $α$ balls of the same colour and $β$ balls of the opposite colour. Both cases, $β=0$ and $β>0$ are well known and correspond respectively to Pólya's and Friedman's replacement schemes. We consider a mixture of both of these: with probability $p\in(0,1]$ balls are replaced according to Friedman's recipe and with probability $1-p$ according to the one by Pólya. Independently of the initial urn composition and independently of $α$, $β$, and the value of $p>0$, we show that the proportion of balls of one colour converges almost surely to $\frac12$. The latter is the limit behaviour obtained by using Friedman's scheme alone, i.e. when $p=1$. Our result follows by adapting an argument due to D. S. Ornstein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Friedman vs Pólya Alves, Raphael Rosales, Rafael A. Probability Primary: 60F15, Secondary: 60G42, 60J05 Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $α$ balls of the same colour and $β$ balls of the opposite colour. Both cases, $β=0$ and $β>0$ are well known and correspond respectively to Pólya's and Friedman's replacement schemes. We consider a mixture of both of these: with probability $p\in(0,1]$ balls are replaced according to Friedman's recipe and with probability $1-p$ according to the one by Pólya. Independently of the initial urn composition and independently of $α$, $β$, and the value of $p>0$, we show that the proportion of balls of one colour converges almost surely to $\frac12$. The latter is the limit behaviour obtained by using Friedman's scheme alone, i.e. when $p=1$. Our result follows by adapting an argument due to D. S. Ornstein. |
| title | Friedman vs Pólya |
| topic | Probability Primary: 60F15, Secondary: 60G42, 60J05 |
| url | https://arxiv.org/abs/2502.16768 |