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!