The Efron-Stein inequality for identically distributed pairs

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 \(\rho(k)\le\max_i k_i/2\).