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It is well-known that if $\xi$ is an absolutely continuous random variable with characteristic function $\phi(t)$, then for each $\epsilon>0$ one has $\sup\limits_{|t|>\epsilon}|\phi(t)|<1$ (sometimes it is called Cramer's theorem). However, if we can say something about existence of moments of the r.v. then this result can be improved. For instance, if $\mathsf E \xi=0,\, \mathsf E \xi^2<\infty$, then one can conclude(if I do not mistake) that exists such $K>0$ that for all $s \in (0,1)$ $\sup_{|t|>s}|\phi(t)|. The questions are the following: is the statement above true? How one can prove it(I tried to do something with Taylor's expansion and the estimation like $|e^{is}-1-is|\leq |s|^2/2$, but I did not manage to prove it)? Can there be generalization for r.v. that possesses higher moments?

Thanks

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