2026/05/31 by Rikhav Shah, Vilas Winstein
Mathematics · #math.PR #msc:60G48 #msc:60G07 #msc:60G10 #acm:60G48 #acm:60G07 #acm:60G10
arxiv created 2026/08/02 · arxiv updated 2026/08/04
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.