Square Root of a Characteristic Function. For any $\phi$ a characteristic function, is there a characteristic function $\Phi$ such that $\Phi ^2=\phi$?
I know that if $\phi$ is a characteristic function, then $\phi ^2$ is as well. 
Also that $\sqrt{|\phi|}$ is not always a characteristic function.
But I don’t know how to get to the other way round.
Thanks!!!

In general, not.  For example, $\phi(t)=\cos(t)$, the characteristic function for an unbiased $\pm 1$ r.v. cannot be written as $\phi=\Phi^2$ for any characteristic function $\Phi$. (For if there were such a $\Phi$, there would be a probability measure $\mu$ for which $\Phi(t)=\int\exp(itx)\mu(dx)$, and for which $\mu*\mu$ was the uniform distribution on the set $\{-1,1\}$.  The support of $\mu$ must be a set $S$ such that $S+S=\{\pm1\}$.  Such an $S$  set must have more than $1$ element, or else the cardinality of $S+S$ is too small.  But if it has two or more elements, the cardinality of $S+S$ is at least $3$.  Therefore there is no such $S$, and hence no such $\mu$, and finally, no such $\Phi$.) 
If $\phi$ is the characteristic function of an infinitely divisible distribution,  then yes, but  this is an exceptionally tiny subset of all characteristic functions.
