 Talagrand's one-sided bounded differences inequality

A useful precise version of Talagrand's one-sided bounded differences inequality assumes that independent random variables $X_i$ enter a measurable function $f$ for which there are nonnegative coordinate weights $c_i(x)$ satisfying the global comparisonPut $Z=f(X)$ and $V=\mathbb{E}{\sum }_i{c}_i(X{)}^2$. When $0<V<\infty$ and $Z$ is integrable,The global comparison is stronger than bounding separate coordinate changes at one input. This distinction matters for a lower tail.^[ref]

For a differentiable convex function on $[0,1{]}^n$, the supporting-gradient inequality verifies the comparison with $c_i(x)=|{\partial }_{i}f(x)|$. Thus $V=\mathbb{E}\Vert \nabla f(X){\Vert }^2$ controls the lower tail. Applying the theorem to the negative of a concave function gives its upper tail. For example, $f(x)={\sum }_i{x}_i$ has $V=n$, giving $e^{-t^2/(2n)}$ as a valid, though conservative, lower-tail bound.