Lower Estimate on Square Function of an Indicator Set

TL;DR

This paper establishes a lower bound for the expected square of a martingale square function over indicator sets, extends the result to wavelet square functions, and discusses related open questions.

Contribution

It introduces a lower estimate for the square function of indicator sets and extends the result to wavelet square functions, addressing a gap in existing martingale theory.

Findings

Lower bound for martingale square function on indicator sets

Extension of the bound to wavelet square functions

Discussion of open questions in the area

Abstract

Let $Sf$ be a discrete martingale square function. Then, for any set $V$ of positive probability, we have $\mathbb{E} S(\mathbf{1}_V)^2 \geq \eta \mathbb{P}(V)$ for an absolute constant $\eta >0$. We extend this to wavelet square functions, and discuss some related open questions.