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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.03206 |
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| _version_ | 1866915993277693952 |
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| author | Shah, Rikhav Winstein, Vilas |
| author_facet | Shah, Rikhav Winstein, Vilas |
| contents | This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution, and we provide a proof that elucidates the connection to two classical arcsine laws for Brownian motion. The notion of a martingale is further generalized, and a larger class of walks is considered and similarly characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The maximum-entropy median-martingale Shah, Rikhav Winstein, Vilas Probability 60G48, 60G07, 60G10 G.3 This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution, and we provide a proof that elucidates the connection to two classical arcsine laws for Brownian motion. The notion of a martingale is further generalized, and a larger class of walks is considered and similarly characterized. |
| title | The maximum-entropy median-martingale |
| topic | Probability 60G48, 60G07, 60G10 G.3 |
| url | https://arxiv.org/abs/2605.03206 |