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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!

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  • $\begingroup$ Schwartz space (?) though it might be too restrictive $\endgroup$ – user17762 May 25 '11 at 1:07
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    $\begingroup$ 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? $\endgroup$ – Giuseppe Negro Jun 7 '11 at 0:19
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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 $$ \lim_{t\to+\infty}\frac1t\log P(|X|\ge t)=-\infty. $$

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