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<p>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&#xa0;<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{BMO}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>BMO</mtext> </math></EquationSource> </InlineEquation>. 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.</p>

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Martingale transforms of bounded random variables and indicator functions of events

  • 期刊论文

  • 发表日期：

    2025年9月10日

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  • Dmitriy Stolyarov,
  • Vasily Vasyunin,
  • Pavel Zatitskii

摘要

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  \(\textrm{BMO}\) 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.