Let $N$ be a Poisson process with intensity $\lambda$, for $0<\sigma<\tau$, we have $$E[\sup_{\sigma\le t\le \tau}(N_t/t-\lambda)^2]\le 4\tau\lambda/\sigma^2$$
I think this is by Doob’s inequality for a martingale, but it works only for a positive martingale, and I don’t know how to handle the sign of $M_t=N_t/t-\lambda$