The maximum-entropy median-martingale on [0,1] has the arcsine distribution as its stationary distribution, with a proof linking it to two classical arcsine laws for Brownian motion, plus a generalization of martingales to a larger class of walks.
Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form
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The maximum-entropy median-martingale
The maximum-entropy median-martingale on [0,1] has the arcsine distribution as its stationary distribution, with a proof linking it to two classical arcsine laws for Brownian motion, plus a generalization of martingales to a larger class of walks.