The maximum-entropy median-martingale
Probability
2026-05-11 v2
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.
Keywords
Cite
@article{arxiv.2605.03206,
title = {The maximum-entropy median-martingale},
author = {Rikhav Shah and Vilas Winstein},
journal= {arXiv preprint arXiv:2605.03206},
year = {2026}
}