Characteristic exponent random variable well-defined?

Question

Let $X: \Omega \to \mathbb{R}$ be a random variable. We define the characteristic exponent of $X$ by

$$\Psi(u) := -\log\mathbb{E}(e^{iuX})$$

Is this well-defined? How are we sure that we don't get something like $-\log(-1)$ (assuming principal branch of logarithm) and get something undefined?

Answer 1

There is a problem when the characteristic function vanishes at some point. Otherwise there is a unique continuous function $\Psi$ such that $\Psi(0)=0$ and $e^{-\Psi (t)}=Ee^{itX}$ for all $t$. For a proof of this see Theorem 7.6.2 in Chung's book [applied to each of the intervals $[-N,N]$.