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Let $X$ be an $\mathbb R$-valued random variable and let $f(t)$ be his characteristic function. If $f$ is derivable up to order $k$ in $t=0$ , where $k$ is even, then $X$ has a finite $k$-th expectation.

I ask for a counterexample where $k$ is not even.

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  • $\begingroup$ Welcome to MSE. Please read this text about how to ask a good question. $\endgroup$ – José Carlos Santos Feb 1 at 9:52
  • $\begingroup$ Have a look at this. $\endgroup$ – drhab Feb 1 at 10:01
  • $\begingroup$ drhab can you give me an hint to prove that the characteristics function is derivable in 0 ? $\endgroup$ – Dario La Torre Feb 1 at 10:55

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