= Janson lower-tail bound by independent thinning
{c}
{title2=$\Pr(X\le\mu-a)\le\exp[-a^2/(2(\mu+\Delta))]$}
For increasing containment indicators, attach independent Bernoulli selectors of parameter $t$. <Janson inequality> applied to the selected events gives $\mathbb E(1-t)^X\le\exp(-t\mu+t^2\Delta/2)$. Set $t=1-e^{-s}$ and apply the <exponential Markov bound>. The estimates $s-s^2/2\le1-e^{-s}\le s$ give exponent at most $-sa+s^2(\mu+\Delta)/2$. Optimizing at $s=a/(\mu+\Delta)$ proves the displayed lower-tail bound, with the dependency sum ordered and excluding the diagonal.
Back to article page
