Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

Abstract

Let $X$ be a (quasi-)Banach space. Let $d=(d_n)_{n\geq1}$ be an $X$-valued sequence of random variables adapted to a filtration $(\mathcal{F}_n)_{n\geq1}$ on a probability space $(\Omega,\mathcal{A},\mathbb{P})$, define $\mathcal{F}_{\infty}:=\sigma(\mathcal{F}_n : n\geq1)$ and let $e=(e_n)_{n\geq1}$ be a $\mathcal{F}_{\infty}$-conditionally independent sequence on $(\Omega,\mathcal{A},\mathbb{P})$ such that $\mathcal{L}(d_n | \mathcal{F}_{n-1}) = \mathcal{L}(e_n | \mathcal{F}_{\infty})$ for all $n\geq1$ ($\mathcal{F}_0=\{\Omega, \varnothing\}$). If there exists a $p\in(0,\infty)$ and a constant $D_p$ independent of $d$ and $e$ such that one has, for all $n\geq1$, $$(^{*})\qquad \mathbb{E}\Biggl \Vert \sum_{k=1}^{n}d_{k}\Biggr \Vert ^{p}\leq D_{p}^{p}\mathbb{E}\Biggl \Vert \sum_{k=1}^{n}e_{k}\Biggr \Vert ^{p},$$ then $X$ is said to satisfy the decoupling inequality for $p$. It has been proven that $X$ is a UMD space if and only if both $(^{*})$ and the reverse estimate hold for some (all) $p\in(1,\infty)$. However, in earlier work we proved that the space $L^1$, which is not a UMD space, satisfies the decoupling inequality for all $p\geq1$.

Here, we prove that if the decoupling inequality is satisfied in $X$ for some $p\in(0,\infty)$ then it is satisfied for all $p\in(0,\infty)$. We consider the behavior of the constant $D_p$ in $(^{*})$. We examine its relation to the norm of the Hilbert transform on $L^p(X)$ and show that if $X$ is a Hilbert space then there exists a universal constant $D$ such that $(^{*})$ holds with $D_p=D$, for all $p\in[1,\infty)$.

An important motivation to study decoupling inequalities is that they play a key role in the recently developed theory for stochastic integration in Banach spaces. We extend the available theory, proving a $p$th-moment Burkholder-Davis-Gundy inequality for the stochastic integral of an $X$-valued process, where $X$ is a UMD space and $p\in(0,\infty)$.