Given a random variable $X$, it's characteristic funciton is defined as:

$$\phi_X(t) = \mathbb{E}[e^{itX}]$$

I'm wondering what conditions are required for the characteristic function of a random variable to be differentiable (i.e. for $\frac{d\phi_X(t)}{dt}$ to exist)?

  • 1
    $\begingroup$ See this for instance. Many textbooks on probability theory should have mentioned similar results. $\endgroup$ – user1551 Nov 28 '12 at 16:28
  • $\begingroup$ This question is fully dealt with at the obvious place. $\endgroup$ – Did Dec 2 '12 at 12:35

One can use the characteristic function of a random variable to find its moments. So if $X$ has $k$ moments ($E|X|^k <\infty \, a.s. $), then the characteristic function with be $k$-times differentiable on the entire real line as one of it's properties is that is uniformly continues on the entire space.

Maybe this will not be helpful, as you most likely will want to know if a random variable has moments looking at the characteristic function.


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