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arXiv · 2605.03206

The maximum-entropy median-martingale

Rikhav Shah·Vilas Winstein
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Abstract

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.

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Rikhav Shah, Vilas Winstein. 2026-08-02. The maximum-entropy median-martingale. https://arxiv.org/abs/2605.03206

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