TITLE: Holomorphic Characteristic Function of a Random Variable
QUESTION [7 upvotes]: Let $X$ be a random variable with distribution $\mu _X$.  Then, we define the characteristic function of $X$, $\phi _X$, by
$$
\phi _X(t)\equiv \mathrm{E}\left[ e^{itX}\right] =\int _\mathbb{R} e^{itx}d\mu _X(x)
$$
This integral always exists for $t\in \mathbb{R}$.  I am trying to determine a "good" set of assumptions to place on $X$ so as to guarantee that this integral makes sense for all $t\in \mathbb{C}$ and so that the resulting function is entire.
I have tried several things, but to no avail.  I fear as if I have not even come up anything worthy of mentioning.  Any thoughts/hints/suggestions/solutions would be most welcome.
Thanks much!

REPLY [5 votes]: Since $X$ is real valued, the property that, for every complex number $z$, the random variable $\mathrm{e}^{zX}$ is integrable is equivalent to the property that
$$
\lim_{t\to+\infty}\frac1t\log P(|X|\ge t)=-\infty.
$$