= Poisson maximal concentration bound
{c}
{title2=$\Pr(\sup_{t\leq L}|N(t)-t|\geq L\eta)\leq2e^{-Lh(\eta)}$}
For a unit-rate <Poisson process>, $h(\eta)=(1+\eta)\log(1+\eta)-\eta$ gives the displayed maximal deviation bound for $L,\eta>0$. <Poisson exponential martingales> and the <Ville inequality> control upper and lower deviations; the upper-tail rate $h$ also bounds the lower-tail probability. Since $h(0)=0$ and $h'(\eta)=\log(1+\eta)>0$, the rate is strictly positive for every positive deviation level.
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