The Efron-Stein inequality for identically distributed pairs

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 $\rho (k)\le {\max }_i{k}_i\mathrm{/}2$.