How can one prove the following inequalities for a standard Poisson process $\mathbf{N}(t)$ ?
$\mathbb{P}\bigg[\bigg|\frac{\mathbf{N}(\lambda)}{\lambda}-1 \bigg| > \varepsilon\bigg] \leq \exp\big(-C_1\varepsilon^2 \lambda\big)$ for a fixed $\lambda$ and, more generally,
$\mathbb{P}\bigg[\sup_\limits{t \geq \Lambda}\bigg|\frac{\mathbf{N}(t)}{t}-1 \bigg| > \varepsilon\bigg] \leq \exp\big(-C_2\varepsilon^2 \Lambda\big)$ ?
If the proof is involved, a reference would be a good idea.