= Chi-squared Chernoff lower-tail bound
{title2=$\mathbb P(V\leq na)\leq e^{-n(a-1-\log a)/2}$}
For $V\sim\chi_n^2$ and $0<a<1$, apply the <Chernoff bound> to $e^{-sV}$ using $\mathbb E e^{-sV}=(1+2s)^{-n/2}$. Optimization at $s=(a^{-1}-1)/2$ proves the displayed estimate. It remains useful when the additive lower threshold in the <chi-squared concentration inequality> is nonpositive. For $0<r\leq1$, choosing $a=e^{-4r}$ gives a tail at most $e^{-nr^2}$ because $e^{-4r}-1+4r\geq2r^2$.
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