Self-normalized processes arise naturally in many statistical tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there is less work on multidimensional processes outside of the sub-Gaussian setting. In this work, we construct a general, self-normalized inequality for $\mathbb{R}^d$-valued processes that satisfy a simple yet broad "sub-$\psi$" tail condition, which generalizes assumptions based on cumulant generating functions. From this general inequality, we derive an upper law of the iterated logarithm for sub-$\psi$ vector-valued processes, which is tight up to small constants. We demonstrate applications in prototypical statistical tasks, such as parameter estimation in online linear regression and auto-regressive modeling, and bounded mean estimation via a new (multivariate) empirical Bernstein concentration inequality.

翻译：自归一化过程自然出现于许多统计任务中。尽管自归一化集中性在标量值过程中已得到广泛研究，但在亚高斯设定之外的多维过程中相关研究较少。本文为满足简单而广泛的"子ψ"尾部条件（该条件推广了基于累积生成函数的假设）的$\mathbb{R}^d$值过程构建了一个通用的自归一化不等式。基于该一般性不等式，我们推导了子ψ向量值过程的迭代对数上界律，该结果在小常数范围内是紧致的。我们通过在线性回归和自回归建模中的参数估计，以及通过新的（多元）经验Bernstein集中不等式实现有界均值估计等典型统计任务，展示了其应用。