Showing that $\exp\big(-|x|^p\big)$ is not a characteristic function

Question

I need to show that $\exp\big(-|x|^p\big)$ is not a characteristic function of a non negative pdf for $p>2$. I am a bit lost as to how to approach this problem.

Thank in advance

Answer 1

Let $\phi (x)=e^{-|x|^{p}}$. Let $X$ be random variable with this characteristic function. It is easy to see that when $p>2$ $\phi ''(0)$ exists and $\phi ''(0)=0$. This implies that $EX^{2}=0$. But then $X=0$ almost surely and $\phi (x)=Ee^{itX}=1$ for all $x$. This contradiction shows that there is no random variable with this characteristic function.