It is shown that functions defined on {0, 1, …, r−1}n satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger “local” sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s [Ann. Probab. 22 (1994) 1576–1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on {0, 1, …, r−1}n for r≥2.
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