Weakly self-bounding function

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 $Z_i=f_i(X^{(i)})$.

For a nonnegative weakly self-bounding function $Z=f(X_1,\dots ,X_n)$ of independent random variables, a Gaussian lower-tail conclusion requires a further coordinate-increment hypothesis in the standard theorem. Suppose $m=\mathbb{E}Z\in (0,\infty )$ and the measurable coordinate-deleted functions satisfyThenThis is the weakly $(a,b)$ self-bounding lower-tail theorem with $a=1,b=0$. The bound on each increment is a separate hypothesis and must not be dropped when invoking that theorem.^[ref]

For example, for $Z={\sum }_i{X}_i$ with independent Bernoulli distribution variables, deleting coordinate $i$ changes $Z$ by $X_i\in \{0,1\}$. Both displayed hypotheses hold since ${\sum }_i{X}_i^2=Z$. With $m=100$, a downward deviation of $20$ has probability at most $e^{-2}$. If $m=0$, nonnegativity gives $Z=0$ almost surely; if $t>m$, the lower-tail event is empty.