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arXiv · 2310.02362

Martingale transforms of bounded random variables and indicator functions of events

Dmitriy Stolyarov·Vasily Vasyunin·Pavel Zatitskii
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Abstract

We provide sharp estimates for the distribution function of a martingale transform of the indicator function of an event. They are formulated in terms of Burkholder functions, which are reduced to the already known Bellman functions for extremal problems on $\mathrm{BMO}$. The reduction implicitly uses an unexpected phenomenon of automatic concavity for those Bellman functions: their concavity in some directions implies concavity with respect to other directions. A similar question for a martingale transform of a bounded random variable is also considered.

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Dmitriy Stolyarov, Vasily Vasyunin, Pavel Zatitskii. 2023-10-03. Martingale transforms of bounded random variables and indicator functions of events. https://arxiv.org/abs/2310.02362

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