Maximal Inequalities for Independent Random Vectors

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.