Limit behavior of Poisson's process near zero

Question

If $N(t)$ is a Poisson process how to say that $\lim\limits_{t\to 0} \frac{N(t)}{t} = 0$ (convergence in probability). Probably I have to use properties of Poisson process, but I have any ideas. Any hints?

Answer 1

You have to show that for each fixed $\epsilon>0$, that $\lim_{t\to0}P(N(t)>\epsilon t)=0$.

But $N(t)$ is Poisson with expectation $t$, and hence non-negative integer valued. So if $t\epsilon<1$, the event $[N(t)>\epsilon t]$ is the complement of the event $[N(t)=0]$, and so $$P(N(t)>\epsilon t)=1-P(N(t)=0)=1-\exp(-t) ,$$ and its limit as $t\to0$ is $0$.