Is it true that $\mathbb{P}(|\xi_t|>C)>0$ for homogeneous process with independent increments?

Question

Let $\xi_t, t \in \mathbb{R}_+$ be a homogeneous non-degenerate stochastic process with independent increments. Is it true that $\mathbb{P}(|\xi_t|>C)>0$ for all $t,C>0$?

Any hints to start with?

Answer 1

Yes. We can write $\xi_t$ as $\xi_0+(\xi_{t/n}-\xi_{0})+...+\xi_t-\xi_{(n-1)t/n})$. From the assumptions it follows that the distribution of $\xi_t$ is the sum of $n$ i.i.d. random variables. It follows that $\xi_t$ has an infinitely divisible distribution. No random variable with an infinitely divisible distribution can be bounded unless it is degenerate. [ You can search MSE for a proof of this].