Title: Maximal Inequalities for Independent Random Vectors Show Chinese title

Title: 独立随机向量的最大不等式

Abstract: Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study of empirical risk minimizers. Although the expected supremum over an infinite collection appears more often in these applications, the expected supremum over a finite collection is a basic building block. This follows from the generic chaining argument. For the case of finite maximum, most existing bounds stem from the Bonferroni inequality (or the union bound). The optimality of such bounds is not obvious, especially in the context of heavy-tailed random vectors. In this article, we consider the problem of finding sharp upper and lower bounds for the expected $L_{\infty}$ norm of the mean of finite-dimensional random vectors under marginal variance bounds and an integrable envelope condition. Show Chinese abstract

Abstract: 极大不等式是指关于随机变量集合上平均值的上确界期望值的界。它们在非参数估计和高维估计的研究中起着至关重要的作用，特别是在经验风险最小化器的研究中。尽管在这些应用中更常见的是无限集合上的期望上确界，但有限集合上的期望上确界是一个基本构建块。这源于通用链式论据。对于有限最大值的情况，大多数现有的界来源于邦弗伦尼不等式（或联合界）。这些界的最优性并不明显，特别是在重尾随机向量的情况下。本文中，我们考虑在边缘方差约束和可积包络条件下的有限维随机向量均值的$L_{\infty}$范数的期望的尖锐上下界的寻找问题。