We present new concentration inequalities for either martingale dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing standard Chernoff bounds to self-normalized heavy-tailed settings. These inequalities are often randomized in a way that renders them strictly tighter than existing deterministic results in the literature, are typically expressed in the Loewner order, and are sometimes valid at arbitrary data-dependent stopping times. Along the way, we explore the theory of matrix supermartingales and maximal inequalities, potentially of independent interest.

翻译：本文针对鞅依赖或可交换的随机对称矩阵，在涵盖标准Chernoff界至自正则化重尾场景的多种尾部条件下，提出了新的集中不等式。这些不等式常通过随机化处理，使其严格优于文献中现有的确定性结果，通常以Loewner序形式表述，且部分结论在任意数据依赖的停时下依然成立。在此过程中，我们探讨了矩阵超鞅与极大不等式理论，这些内容可能具有独立的研究价值。