Explain what is meant by the characteristic function $\phi$ of a real-valued random variable and prove that $|\phi|^{2}$ is also a characteristic function of some random variable.

Let us say that a characteristic function $\phi$ is infinitely divisible when, for each $n \geqslant 1$, we can write $\phi=\left(\phi_{n}\right)^{n}$ for some characteristic function $\phi_{n}$. Prove that, in this case, the limit

$$\psi(t)=\lim _{n \rightarrow \infty}\left|\phi_{2 n}(t)\right|^{2}$$

exists for all real $t$ and is continuous at $t=0$.

Using Lévy's continuity theorem for characteristic functions, which you should state carefully, deduce that $\psi$ is a characteristic function. Hence show that, if $\phi$ is infinitely divisible, then $\phi(t)$ cannot vanish for any real $t$.