Doob's inequality for non-commutative martingales

TL;DR

This paper extends Doob's inequality to non-commutative martingales in $L_p$ spaces, providing bounds for sums of conditional expectations and establishing a non-commutative maximal inequality.

Contribution

It generalizes classical Doob's inequality to the non-commutative setting, using duality and operator algebra techniques.

Findings

Established $L_p$ bounds for sums of conditional expectations in non-commutative spaces.

Derived a non-commutative version of Doob's maximal inequality for $1<p extless\8$.

Extended classical martingale inequalities to the non-commutative framework.

Abstract

Let $1\le p<\8$ and $(x_n)_{\nen}$ be a sequence of positive elements in a non-commutative $L_p$ space and $(E_n)_{\nen}$ be an increasing sequence of conditional expectations, then the $L_p$ norm of \sum_n E_n(x_n) can be estimated by c_p times the $L_p$ norm of \sum_n x_n. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for $1<p\le \8$.