We present new concentration inequalities for either martingale dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-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 positive semidefinite supermartingales and maximal inequalities, a natural matrix analog of scalar nonnegative supermartingales that is potentially of independent interest.

翻译：我们针对鞅依赖或可交换随机对称矩阵，在多种尾部分布条件下（涵盖从标准Chernoff界到自归一化重尾情形）提出了新的浓度不等式。这些不等式通常以随机化形式呈现，使其严格优于文献中现有的确定性结果，常以Loewner偏序表述，且有时在任意依赖于数据的停时下仍成立。在此过程中，我们探讨了正定半定超鞅与极大不等式理论——作为标量非负超鞅的自然矩阵类比，该理论本身可能具有独立研究价值。