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For a unit-rate Poisson process and real $\theta$, the displayed process is a nonnegative martingale with initial value one. Independent increments give $\mathbb{E}{e}^{\theta (N_t-N_s)}=\exp ((t-s)(e^{\theta }-1))$, proving its conditional mean property. The Ville inequality turns this exponential moment into a maximal tail bound.

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.