Self-bounding function

A nonnegative function $f$ of independent coordinates is self-bounding when there are coordinate-deleted versions $f_i$ such that $0\le f-f_i\le 1$ andThese inequalities make the natural variance proxy no larger than the function itself.

A nonnegative function $f$ is weakly self-bounding when coordinate-deleted functions $f_i$ satisfy $f_i\le f$ andThe squared coordinate sensitivity is then controlled by the observed value of the function itself.

For a weakly self-bounding function $Z=f(X_1,\dots ,X_n)$ of independent random variables, the coordinatewise variance inequality giveswhere $Z_i=f_i(X^{(i)})$.

The entropy method givesfor $\lambda \ge 0$. Consequently the Chernoff bound yields $\mathbb{P}(Z-\mathbb{E}Z\le -t)\le \exp [-t^2/(2\mathbb{E}Z)]$.