Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exponential Concentration Inequalities For Independent Random Vectors Under Sublinear Expectations

Nahom Seyoum

Abstract

Li and Hu recently established variance-type O(1/n) bounds for the sample mean of independent random vectors under sublinear expectations. We extend their results to the exponential concentration regime. For bounded, independent R^d-valued random vectors under a regular sublinear expectation, we prove: (i) a general concentration principle that reduces vector-valued tail bounds to scalar martingale inequalities via a three-layer architecture; (ii) an Azuma-Hoeffding inequality showing that the distance from the sample mean to the Minkowski average of the expectation sets has sub-Gaussian tails; (iii) a Bernstein inequality incorporating the variance parameter of Li and Hu, interpolating between sub-Gaussian and sub-exponential regimes; (iv) a dimension-free bound replacing the exponential covering prefactor with a polynomial one via the matrix Freedman inequality; and (v) an explicit construction demonstrating that the sub-Gaussian rate is optimal. To the best of our knowledge, these constitute the first exponential concentration inequalities for the multivariate sample mean under sublinear expectations in terms of the set-valued distance to the Minkowski average.

Exponential Concentration Inequalities For Independent Random Vectors Under Sublinear Expectations

Abstract

Li and Hu recently established variance-type O(1/n) bounds for the sample mean of independent random vectors under sublinear expectations. We extend their results to the exponential concentration regime. For bounded, independent R^d-valued random vectors under a regular sublinear expectation, we prove: (i) a general concentration principle that reduces vector-valued tail bounds to scalar martingale inequalities via a three-layer architecture; (ii) an Azuma-Hoeffding inequality showing that the distance from the sample mean to the Minkowski average of the expectation sets has sub-Gaussian tails; (iii) a Bernstein inequality incorporating the variance parameter of Li and Hu, interpolating between sub-Gaussian and sub-exponential regimes; (iv) a dimension-free bound replacing the exponential covering prefactor with a polynomial one via the matrix Freedman inequality; and (v) an explicit construction demonstrating that the sub-Gaussian rate is optimal. To the best of our knowledge, these constitute the first exponential concentration inequalities for the multivariate sample mean under sublinear expectations in terms of the set-valued distance to the Minkowski average.
Paper Structure (9 sections, 14 theorems, 40 equations)

This paper contains 9 sections, 14 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Preliminaries
  5. Martingale reduction
  6. Concentration via scalar tail bounds
  7. Dimension-free bound for identically distributed vectors
  8. Optimality of the sub-Gaussian rate (up to constants)
  9. Discussion and open problems

Key Result

Theorem 2.1

There exists a convex and weakly compact set of probability measures $\mathcal{P}$ on $(\Omega, \mathcal{B}(\Omega))$ such that eq:representation holds. The associated upper capacity is $\hat{V}(A) := \sup_{P \in \mathcal{P}} P(A)$.

Theorems & Definitions (33)

  • Theorem 2.1: Representation DenisHuPeng2011HuPeng2009
  • Definition 2.2: Independence Peng2019
  • Proposition 2.3: HuLiLi2021
  • Theorem 2.4: LiHu2024
  • Lemma 3.1: Conditional domination
  • proof
  • Remark 3.2: Role of the truncation
  • Lemma 3.3: Martingale reduction
  • proof
  • Lemma 3.4: Covering transfer
  • ...and 23 more