Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

TL;DR

This paper characterizes the probability space structures that determine the martingale type of Banach spaces, focusing on intrinsic filtration conditions that influence upper $ ext{ell}^p$ estimates and their implications.

Contribution

It identifies intrinsic conditions on filtrations of atomic $\sigma$-algebras that ensure Banach spaces have a specific martingale type, extending prior foundational work.

Findings

Characterizes probability spaces determining martingale type

Identifies intrinsic filtration conditions for $ ext{ell}^p$ estimates

Extends and complements prior martingale type research

Abstract

In the present paper we identify those filtered probability spaces $(\Omega,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $\sigma$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(\Omega,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(\Omega,\, X)}+\sum_{n=1}^{\infty} \|\Delta_n f\|^p_{L^p(\Omega,\, X)}\right),\qquad f\in L^p(\Omega,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.