On decoupling in Banach spaces

TL;DR

This paper investigates decoupling inequalities for Banach space-valued random variables, showing invariance under filtration enlargement and applications to moment inequalities and stochastic integrals.

Contribution

It introduces a framework restricting conditional distributions in decoupling, allowing approximation by Haar expansions and establishing invariance properties.

Findings

Decoupling constants for b^d with l^-norm are established.

Decoupling inequalities lead to Burkholder-Davis-Gundy type inequalities for Banach space-valued stochastic integrals.

Invariance of decoupling properties under filtration enlargement is proven.

Abstract

We consider decoupling inequalities for random variables taking values in a Banach space $X$. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar type expansion in which only the same conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not effect the decoupling properties (e.g., the constants involved). As special case we deal with one-sided moment inequalities when decoupling dyadic (i.e., Paley-Walsh) martingales. We establish the decoupling constant of $\mathbb{R}^d$ with the $l^{\infty}$-norm. As an example of an application, we demonstrate that Burkholder-Davis-Gundy type inequalities for stochastic integrals of $X$-valued processes can be obtained from decoupling inequalities for $X$-valued dyadic martingales.