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arXiv · 2605.05388

The Efron-Stein inequality for identically distributed pairs

Jnaneshwar Baslingker·Bálint Virág
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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 \(\rho(k)\le\max_i k_i/2\).

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Jnaneshwar Baslingker, Bálint Virág. 2026-05-06. The Efron-Stein inequality for identically distributed pairs. https://arxiv.org/abs/2605.05388

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