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A matrix Burkholder-Davis-Gundy inequality

Tom Maitre

TL;DR

This work develops a BDG-type inequality for the spectral norm of matrix-valued stochastic integrals, bridging non-commutative Khintchine and Burkholder–Davis–Gundy inequalities. By establishing Schatten-norm bounds via Itô calculus and proving a Freedman-type matrix inequality, the authors bound the moments of the supremum of $\|X_t\|$ in terms of the quadratic variation $\langle X\rangle_t$, up to a dimension-dependent $\sqrt{\log n}$ correction. The results are extended to rectangular matrices via Hermitian dilation, with explicit constants $C=2\sqrt{2}$, and the approach clarifies the role of noncommutativity and dimension in stochastic matrix analysis. Overall, the paper provides a concrete, operator-valued generalization of BDG that is sharp up to the log-dimension term and adaptable to broader matrix settings.

Abstract

We prove an inequality for the spectral norm of matrix valued stochastic integrals. This inequality can be seen either as a non-commutative version of the Burkholder-Davis-Gundy inequality or as an extension of the non-commutative Khintchine inequality of Lust-Piquard to stochastic integrals. The proof relies on a version of Freedman's inequality for matrix valued martingales.

A matrix Burkholder-Davis-Gundy inequality

TL;DR

This work develops a BDG-type inequality for the spectral norm of matrix-valued stochastic integrals, bridging non-commutative Khintchine and Burkholder–Davis–Gundy inequalities. By establishing Schatten-norm bounds via Itô calculus and proving a Freedman-type matrix inequality, the authors bound the moments of the supremum of in terms of the quadratic variation , up to a dimension-dependent correction. The results are extended to rectangular matrices via Hermitian dilation, with explicit constants , and the approach clarifies the role of noncommutativity and dimension in stochastic matrix analysis. Overall, the paper provides a concrete, operator-valued generalization of BDG that is sharp up to the log-dimension term and adaptable to broader matrix settings.

Abstract

We prove an inequality for the spectral norm of matrix valued stochastic integrals. This inequality can be seen either as a non-commutative version of the Burkholder-Davis-Gundy inequality or as an extension of the non-commutative Khintchine inequality of Lust-Piquard to stochastic integrals. The proof relies on a version of Freedman's inequality for matrix valued martingales.
Paper Structure (4 sections, 10 theorems, 55 equations)

This paper contains 4 sections, 10 theorems, 55 equations.

Table of Contents

  1. Introduction and preliminaries
  2. A bound for the Schatten norms
  3. The matrix Freedman inequality
  4. A Burkholder-Davis-Gundy type inequality for the spectral norm

Key Result

Theorem 1.1

Let $(X_{t})$ be a stochastic process of the form (eq_Xt). Then, there exists a universal constant $C$ such that, for all $p \in \mathbb{N}^{*}$ and all $t \in \mathbb{R}_{+}^{*},$ Our proof gives $C = 2 \sqrt 2$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2: Freedmann's matrix inequality
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['lemma1']}
  • Remark 1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 7 more