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Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion the case when $k$ is odd? For example, if we only know that $f$ has its first order derivative, do we know that $\mu$ has finite first moment?

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A good reference regarding this type of results is the book by Eugène Lukacs "Characteristic Function". For example, chapter 2.3 "Characteristic functions and moments" provides results in this direction. Theorems 2.3.1, 2.3.2 and 2.3.3 might be what you are looking for.

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    $\begingroup$ To complement my post and to counter the down vote: in the same chapter there is an example due to Zygmund of a characteristic function with a continuous first derivative but for which the associated law has an infinite first moment... $\endgroup$ – user69642 Jun 13 '20 at 15:36
  • $\begingroup$ Thanks. I saw the example. But it was provided without proof that it is differentiable. Did Zygmund prove it? Can you give a reference to the proof of differentiability? $\endgroup$ – trisct Jun 13 '20 at 15:56
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    $\begingroup$ I think it was stated by Zygmund in this paper projecteuclid.org/euclid.aoms/1177730443 but without proof. For a proof, you should have a look at his fundamental work on trigonometric series. $\endgroup$ – user69642 Jun 13 '20 at 22:11

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