Lemma: Bounded-score concentration

For independent $Z_r\in [-1,1]$,

For independent Bernoulli variables the exponent improves to $2e^{-2n{u}^2}$.

Proof. For a bounded variable on an interval of length $b-a$, let $f(t)=\log \mathbb{E}{e}^{t(Z-\mathbb{E}Z)}$. Under exponential tilting, $f^{\prime \prime }(t)$ is a variance and is at most $(b-a{)}^2/4$; this follows by centering at the interval midpoint. Since $f(0)=f^{\prime }(0)=0$, $f(t)\le t^2(b-a{)}^2/8$. Multiply the moment bounds for independent variables, apply exponential Markov inequality, and optimize over $t$. Apply the same argument to the lower tail.