We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative supermartingales and backward nonnegative submartingales, whose convergence and maximal inequalities are the theoretical foundations for a wide and ever-growing body of results in statistics, econometrics, and theoretical computer science. Our results lead to 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. Further, these inequalities are usually expressed in the Loewner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.

翻译：本文探讨了正定矩阵空间中超鞅与后向次鞅的渐近收敛性及非渐近极大不等式。这些是标量非负超鞅与后向非负次鞅的自然矩阵类比，其收敛性与极大不等式构成了统计学、计量经济学和理论计算机科学中广泛且不断增长的理论成果的基石。我们的研究导出了一系列在多种尾部条件下适用于鞅相依或可交换随机对称矩阵的新集中不等式，涵盖了从现行标准的切尔诺夫界到自归一化重尾设置的多种情形。此外，这些不等式通常以勒夫纳序形式表述，有时对所有样本量或任意数据依赖的停时同时成立，并且常可通过外部随机化因子进行加强。