Local tail bounds for functions of independent random variables
Luc Devroye and Gábor Lugosi
Annals of Probability Volume to appear, pp. 00-00, 2006.

## Abstract

It is shown that functions defined on \$\{0,1,\ldots,r-1\}^n\$ satisfying certain conditions of bounded differences that guarantee subgaussian tail behavior also satisfy a much stronger ``local'' subgaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's (1994) variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on \$\{0,1,\ldots,r-1\}^n\$ for \$r\ge 2\$.