This question was raised to me, and I might need some help with this.
"If $X$ is bounded, meaning $\exists K$, that $|(X)| \le K$ than the characteristic function of $X$ can be derivated infinite times. "
First, I get that if the n-th moment exist, then the characteristic function can be derived n times. So i believe, that if all the moments exist (meaning they are not infinite), then the characteristic function can be derivated infinite times.
However, I can't prove that for a bounded distibution all moments exist. Wikipedia says that "For a distribution of mass or probability on a bounded intervall, the collection of all the moments (of all orders, from 0 to ∞) uniquely determines the distribution (Hausdorff moment problem)", but I cant see whether this means all the moments exist.
Thank you for your help!
if by your problem u mean K is a real number then X indeed has all finite moments and E(abs(X^n))<=K^n