Poisson maximal concentration bound ($\mathrm{Pr}({\sup }_{t\le L}|N(t)-t|\ge L\eta )\le 2e^{-Lh(\eta )}$)

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 $h^{\prime }(\eta )=\log (1+\eta )>0$, the rate is strictly positive for every positive deviation level.