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The Efron-Stein inequality for identically distributed pairs
Jnaneshwar Baslingker, Bálint Virág · 2026-05-07 · via math.PR updates on arXiv.org

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 \(ρ(k)\le\max_i k_i/2\).

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