The Efron-Stein inequality for identically distributed pairs

TL;DR

This paper extends the Efron--Stein inequality to independent exchangeable pairs, providing bounds and counterexamples for i.i.d. pairs, with explicit constants when variables are discrete.

Contribution

It proves the inequality for exchangeable pairs and clarifies its limitations for i.i.d. pairs, offering bounds with explicit constants for discrete variables.

Findings

Efron--Stein inequality holds for independent exchangeable pairs.

The inequality fails for i.i.d. pairs, with a sharp trivial bound.

Explicit bounds are provided for variables taking limited values.

Abstract

We prove that the classical Efron--Stein inequality holds for independent exchangeable pairs \((X_i,Y_i)\). The same inequality fails for independent identically distributed pairs; a simple trigonometric counterexample shows that the trivial Cauchy--Schwarz bound of factor \(n\) is sharp. When each random variable takes at most \(k_i\) values, a useful bound still holds with explicit constant \(\rho(k)\le\max_i k_i/2\).