[Paper Review] Universal concentration for sums under arbitrary dependence

Cosme Louart, Sicheng Tan|arXiv (Cornell University)|Jan 7, 2026

Probability and Risk Models0 citations

TL;DR

The paper proves a universal, asymptotically optimal concentration bound for sums of identically distributed random variables under arbitrary dependence, via an operator framework based on maximally nonincreasing set-valued operators and the subadditivity of expected shortfall.

ABSTRACT

We present a universal concentration bound for sums of random variables under arbitrary dependence, and we prove that it is asymptotically optimal for broad families of marginals admitting a uniform integrable tail-quantile envelope. The bound follows directly from the subadditivity of expected shortfall, a property well known in the risk-measure literature. Our sharpness result relies on an explicit construction of asymptotically extremal couplings. We furthermore provide practical sufficient conditions -- based on convex transformation order comparisons with exponential and power-law envelopes -- under which the bound admits simple, explicit tail profiles.

Motivation & Objective

• Develop a universal tail bound for sums of random variables with arbitrary dependence given marginal laws.
• Introduce an operator-based framework (maximally nonincreasing operators) to encode concentration bounds.
• Show the bound is asymptotically sharp and characterize worst-case dependency profiles.
• Relate the concentration bound to subadditivity of expected shortfall (CVaR) and Hardy transforms.

Proposed method

• Encode concentration as inequalities between maximally nonincreasing operators in M_downarrow.
• Use survival and tail quantile operators S_X and T_X with operator inversion S_X arrow T_X to relate bounds.
• Derive the sharp bound S_{X_1+...+X_n}  (T_X1 + ... + T_Xn) via the Hardy transform: S_{X_1+...+X_n}  (H(T_{X_1}) + ... + H(T_{X_n}))^{-1}.
• Show identically distributed case yields S_{rac{1}{n}1 sum X_k}  H(T_mu)^{-1} independent of n.
• Construct worst-case dependency profiles (slot-variable mixing) to prove asymptotic sharpness (Theorem 2).
• Provide corollaries giving explicit envelopes (e.g., S_mu  C Id^{-q}, S_mu  C E_1) via Hardy transform identities.

Experimental results

Research questions

• RQ1 What universal tail bound can be established for sums of random variables with arbitrary dependence given fixed marginals?
• RQ2 Can the bound be expressed in a way that is independent of the number of summands n for i.i.d. cases?
• RQ3 How sharp is the concentration bound asymptotically, and what are the worst-case dependency structures achieving sharpness?
• RQ4 How can the bound be operationalized using tractable envelopes for the survival function?
• RQ5 What is the relationship between the bound and classical risk measures like expected shortfall (CVaR) through subadditivity?

Key findings

• A universal, asymptotically optimal tail bound is obtained: S_{(X_1+...+X_n)/n}  (H(T_X) )^{-1} in the i.i.d. case (Theorem 1).
• The bound does not depend on n in the i.i.d. setting, yielding S_{rac{1}{n} sum X_k}  mu tail profile via H(T_mu), removing dependence on n.
• A constructive approach (slot variables) shows the bound is sharp: there exist sequences of identically distributed X_i with the same marginal law achieving S_{rac{1}{n} sum X_k}(t) converging to the limiting profile S_{mu,p}.
• Corollaries provide explicit, tractable envelopes for practical use, including S_mu  C Id^{-q} and S_mu  C E_1, via Hardy transform identities and convexity arguments.
• The work connects the bound to subadditivity of expected shortfall (CVaR) and shows how this concept underpins the universal tail bound.
• Theorem 3 shows a link between convexity properties of f  Id^{-q} and the exponential envelope E_1, showing asymptotic equivalence of power-type and exponential-type bounds as q f
  dinfty.