Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal Inequalities for Independent Random Vectors

Supratik Basu, Arun K Kuchibhotla

TL;DR

This work derives sharp, universal constants for the expected $L_{\infty}$ norm of the mean of independent random vectors under marginal-variance and envelope constraints, reducing complex, finite-collection maximal inequalities to tractable IID and bounded-envelope problems. Leveraging Bennett-type inequalities and worst-case Bernoulli constructions, it yields matching upper and lower bounds for the bounded-envelope case and extends to integrable envelopes via a truncation-and-moments framework that connects to Fuk–Nagaev-type results. The results illuminate the precise dependence on dimension $p$, sample size $n$, and envelope parameters $\sigma,B$, clarifying when the maximal inequality is controlled by the tail or by the variance, and they offer improved bounds over classical Bernstein-type approaches, especially under heavy tails. The findings have direct implications for empirical process theory and high-dimensional risk minimization, enabling tighter control of supremums over finite or infinite function classes and guiding the design of sharp, finite-sample estimators. The work also situates its results in relation to existing literature by comparing to Blancheard et al. and showing consistency with i.i.d. Bernoulli-component bounds, while outlining directions for future work on lower bounds in unbounded settings and broader applications in empirical processes.

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.

Maximal Inequalities for Independent Random Vectors

TL;DR

This work derives sharp, universal constants for the expected norm of the mean of independent random vectors under marginal-variance and envelope constraints, reducing complex, finite-collection maximal inequalities to tractable IID and bounded-envelope problems. Leveraging Bennett-type inequalities and worst-case Bernoulli constructions, it yields matching upper and lower bounds for the bounded-envelope case and extends to integrable envelopes via a truncation-and-moments framework that connects to Fuk–Nagaev-type results. The results illuminate the precise dependence on dimension , sample size , and envelope parameters , clarifying when the maximal inequality is controlled by the tail or by the variance, and they offer improved bounds over classical Bernstein-type approaches, especially under heavy tails. The findings have direct implications for empirical process theory and high-dimensional risk minimization, enabling tighter control of supremums over finite or infinite function classes and guiding the design of sharp, finite-sample estimators. The work also situates its results in relation to existing literature by comparing to Blancheard et al. and showing consistency with i.i.d. Bernoulli-component bounds, while outlining directions for future work on lower bounds in unbounded settings and broader applications in empirical processes.

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 norm of the mean of finite-dimensional random vectors under marginal variance bounds and an integrable envelope condition.

Paper Structure

This paper contains 26 sections, 35 theorems, 370 equations.

Table of Contents

  1. Introduction and Motivation
  2. Problem Description
  3. Optimal Bounds under Bounded Envelope
  4. Analysis of the "Bennett Bound"
  5. Analysis for Random Vectors with Bernoulli components
  6. Optimality of the bounds on $\mathcal{E}_{\infty}(\sigma,B)$ for the general case
  7. A comparitive study with blanchard2024tight
  8. Bounds under Integrable Envelope
  9. Properties of the bound
  10. Conclusions and Future Work
  11. Proofs of Results in Section 1
  12. Proof of Theorem \ref{['thm:non-iid-to-iid-reduction']}
  13. Proof of Theorem \ref{['thm:reduction-to-bounded']}
  14. Proof of Lemma \ref{['lem:M-q-sig-B']}
  15. Proofs of Results in Section 2
  16. ...and 11 more sections

Key Result

Theorem 2.1

For any $n, p \ge 1, 0 \le \sigma \le B$,

Theorems & Definitions (57)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Theorem 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.1
  • Theorem 3.4
  • ...and 47 more