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Sharp concentration inequality for the sum of random variables

Cosme Louart, Sicheng Tan

TL;DR

This work tackles concentration inequalities for sums of random variables under arbitrary dependence with fixed marginals. It develops an operator framework based on survival and tail-quantile operators and uses the Hardy transform to derive a universal bound, yielding $S_{\frac{1}{n}\sum X_k} \le \mathcal{H}(T_\mu)^{-1}$ in the iid case. A key contribution is proving asymptotic sharpness of this bound via a constructive dependency mechanism employing mixed slot variables, which shows the bound is tight for large $n$ and characterizes the limiting tail profile $S_{\mu,p}$. Additionally, the paper provides practical envelopes and a convexity-based comparison principle that translate these results into explicit bounds (e.g., $S_{\frac{1}{n}\sum X_k} \le C\left(\frac{q}{q-1}\right)^q \mathrm{Id}^{-q}$ or $S_{\frac{1}{n}\sum X_k} \le C e \mathcal{E}_1$) for common tail shapes, bridging theory and applications.

Abstract

We present a universal concentration bound for sums of random variables under arbitrary dependence, and we prove it is asymptotically optimal for every fixed common marginal law. The concentration bound is a direct - yet previously unnoticed - consequence of the subadditivity of expected shortfall, a property well known to financial statisticians. The sharpness result is a significant contribution relying on the construction of worst-case dependency profiles between identically distributed random variables.

Sharp concentration inequality for the sum of random variables

TL;DR

This work tackles concentration inequalities for sums of random variables under arbitrary dependence with fixed marginals. It develops an operator framework based on survival and tail-quantile operators and uses the Hardy transform to derive a universal bound, yielding in the iid case. A key contribution is proving asymptotic sharpness of this bound via a constructive dependency mechanism employing mixed slot variables, which shows the bound is tight for large and characterizes the limiting tail profile . Additionally, the paper provides practical envelopes and a convexity-based comparison principle that translate these results into explicit bounds (e.g., or ) for common tail shapes, bridging theory and applications.

Abstract

We present a universal concentration bound for sums of random variables under arbitrary dependence, and we prove it is asymptotically optimal for every fixed common marginal law. The concentration bound is a direct - yet previously unnoticed - consequence of the subadditivity of expected shortfall, a property well known to financial statisticians. The sharpness result is a significant contribution relying on the construction of worst-case dependency profiles between identically distributed random variables.
Paper Structure (3 sections, 10 theorems, 65 equations, 2 figures)

This paper contains 3 sections, 10 theorems, 65 equations, 2 figures.

Table of Contents

  1. Maximally nonincreasing operator framework and main results
  2. Mixed slot variables to sharpnen sum concentration tail
  3. Classical sum concentration profiles

Key Result

Theorem 1

Given $n$ random variables $X_1,\ldots, X_n$ admitting expectations, one can bound:

Figures (2)

  • Figure 1: Representation of $T_X$, $\mathrm{Incr}_{\mathbb E[X]}$, $\mathcal{H}(T_X)$ and $S_{\mu,p}^{-1}$ and the asymptotic contact point.
  • Figure 2: (Left) detail of the choice of of the different parameters for a given tail quantile operator and (Right) cyclic mixing of slot variables: value of $X_1,\ldots, X_n, \sum_{k=1}^n X_k$ depending on the values of $\epsilon_1,\ldots, \epsilon_n, \epsilon_1',\ldots, \epsilon_n'$.

Theorems & Definitions (21)

  • Definition 1: Interval Order and Point-wise resolvent Order between operators
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Lemma 1
  • proof
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 11 more