Differences Between Characteristic Function and Moment-Generating Function

Question

Why can't the moment generating function be defined for all random variables, while the characteristic function can be defined for all random variables?

Answer 1

Note that $e^x$ is unbounded for real $x$, but bounded to the border $\{z\in\Bbb C||z|=1\}$ of the convex set $\{z\in\Bbb C||z|\le1\}$ for imaginary $x$. Thus a real variable's MGF isn't necessarily bounded, but its characteristic function has modulus $\le1$.

Answer 2

A characteristic function is almost the same as a moment generating function (MGF), and in fact, they use the same symbol φ — which can be confusing. Furthermore, the difference is that the “t” in the MGF definition $E(e^{tx})$ is replaced by “it”. In other words, the imaginary number is not present in the definition of an MGF. Therefore, the characteristic function has the advantage that it always exists — even when there is no MGF.