Martingale transforms of bounded random variables and indicator functions of events
Classical Analysis and ODEs
2023-10-05 v1 Analysis of PDEs
Probability
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 . 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.
Keywords
Cite
@article{arxiv.2310.02362,
title = {Martingale transforms of bounded random variables and indicator functions of events},
author = {Dmitriy Stolyarov and Vasily Vasyunin and Pavel Zatitskii},
journal= {arXiv preprint arXiv:2310.02362},
year = {2023}
}
Comments
37 pages, 9 figures