Let $N(t)$ be a rate $\lambda$ Poisson process and let $Y_i$ be i.i.d random variables independent of $N$. Define the compound poisson process by $$Z(t):=\sum_{i=1}^{N(t)} Y_i$$ for $t \in [0,1]$.
I want to show
$Z( \cdot )$ is almost surely continuous at a fixed $t \in [0,1]$.
I already showed that the ch.f. of $Z$ is $\phi_Z(u)=e^{{\lambda t} (\phi (u)-1)}$ where $\phi(u)$ is ch.f. of $Y_1$ and that $Z$ has stationary independent increments.
But, I am not sure how to proceed since I don't have cadlag paths or such.
Any help is appreciated.