Martingale transforms of bounded random variables and indicator functions of events

TL;DR

This paper derives precise bounds for the distribution of martingale transforms of indicator functions, utilizing Bellman functions and revealing an unexpected automatic concavity phenomenon.

Contribution

It introduces sharp estimates for martingale transforms of indicator functions using Bellman functions and uncovers an automatic concavity property of these functions.

Findings

Sharp distribution function estimates for martingale transforms of indicator functions

Identification of an automatic concavity phenomenon in Bellman functions

Extension of results to martingale transforms of bounded random variables

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.