Some remarks on tangent martingale difference sequences in $L^1$-spaces

TL;DR

This paper investigates tangent martingale difference sequences in Banach spaces, showing that certain inequalities hold in a broader class of spaces than UMD spaces, including L^1, with implications for decoupling inequalities.

Contribution

It demonstrates that the inequality (*) holds in a larger class of Banach spaces than UMD spaces when only g satisfies the (CI) condition, including L^1.

Findings

The class of spaces satisfying (*) with only g meeting (CI) is broader than UMD spaces.

L^1 space satisfies the inequality under the specified conditions.

Several open problems related to (*) and decoupling inequalities are proposed.

Abstract

Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has \[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L^1. We state several problems related to (*) and other decoupling inequalities.