On the local behavior of processes with independent increments

Let $\xi(t)$, $t\ge0$, $\xi(0)=0$, be a homogeneous process with independent increments. In [2] it was shown that $\lim\limits_{t\to0}(\xi(t)/t)$ exists and is finite if sample functions of $\xi(t)$ have a bounded variation. We prove that, in the opposite case,
$$ \varlimsup_{t\to0}\frac{\xi(t)}t=\infty. $$