The maximum-entropy median-martingale
Rikhav Shah, Vilas Winstein
TL;DR
This paper investigates a maximum-entropy median-martingale walk on the unit interval, revealing its stationary distribution as the arcsine distribution and connecting it to classical Brownian motion laws.
Contribution
It introduces a maximum-entropy median-martingale framework, characterizes its stationary distribution, and generalizes the martingale concept to a broader class of walks.
Findings
Stationary distribution of the walk is the arcsine distribution.
Connection established between the walk and classical arcsine laws for Brownian motion.
Generalization of the martingale concept to a larger class of walks.
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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