# Sufficient Conditions for the Characteristic Function to Be Differentiable

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)?

• See this for instance. Many textbooks on probability theory should have mentioned similar results. – user1551 Nov 28 '12 at 16:28
• This question is fully dealt with at the obvious place. – 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.