Weighted Davis inequalities for martingale square functions

TL;DR

This paper establishes a sharp weighted Davis inequality for Hilbert space valued martingales, providing new bounds involving martingale square functions and extending results to uniformly convex Banach spaces.

Contribution

The paper introduces a novel weighted Davis inequality for martingale square functions, including a sharp bound and a variant for Banach space valued martingales.

Findings

The inequality is sharp and optimal.

It implies several new results about martingale square functions.

Extension to uniformly convex Banach spaces is achieved.

Abstract

For a Hilbert space valued martingale $(f_n)$ and an adapted sequence of positive random variables $(w_n)$, we show the weighted Davis type inequality \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n \Bigr) \leq \mathbb{E} ( f^*_N w^*_N). \] This inequality is sharp and implies several results about the martingale square function. We also obtain a variant of this inequality for martingales with values in uniformly convex Banach spaces.