Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
Schwartz space (?) though it might be too restrictive – user17762 May 25 '11 at 1:07
For absolutely continuous $X$ with pdf $f$, a good condition is exponential decay at infinity: if $\lvert f(x) \rvert \le e^{-\lambda \lvert x \rvert}$ for a $\lambda >0$, then $\Phi_X$ is holomorphic in the strip $\{z \in \mathbb{C} \mid \lvert \Im m(z)\rvert < \lambda\}$. Did you know that? – Giuseppe Negro Jun 7 '11 at 0:19

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 $$ \lim_{t\to+\infty}\frac1t\log P(|X|\ge t)=-\infty. $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.