Universal concentration for sums under arbitrary dependence

TL;DR

This paper introduces a universal concentration inequality for sums of dependent random variables, demonstrating its optimality and providing practical conditions for explicit tail bounds based on risk measure properties.

Contribution

It establishes a new universal concentration bound for dependent variables, leveraging expected shortfall subadditivity, and offers explicit tail profile conditions for practical applications.

Findings

Bound is asymptotically optimal for broad marginals.

Explicit construction of extremal couplings demonstrates sharpness.

Provides practical conditions for explicit tail bounds.

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.