A Banach space is UMD if and only if tail inequalities hold for maximal functions of its tangent conditionally symmetric processes.
Protter.Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability
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Characterizations of the UMD property via tail estimates for tangent processes
A Banach space is UMD if and only if tail inequalities hold for maximal functions of its tangent conditionally symmetric processes.