Can anything be learned about a probability distribution *directly* from its characteristic function? Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us anything about the underlying random variable.
As an example: The moment generating function can be directly used in Chernoff's Lemma:
$$P(X\geq0)\leq MGF_{X}(t),\; \forall t$$
Is there anything that can be similarly done with a Characteristic Function? Does it's complex modulus or absolute square have any meaning? Can we directly interpret its real or complex parts separately?
From the little I have used the Characteristic Function, it seems to be used for showing convergence in distribution, and relies simply upon recognizing the forms of various characteristic equations. However, this narrow view is likely just from inexperience. I would be happy to have my view expanded :-)

We can derive from the inversion formula the bound 
$$\mathbb P\{ |X|\geqslant 2/\delta \}\leqslant \delta^{-1}\int_{-\delta}^\delta (1-\varphi_X(s))\mathrm ds.$$
