Let $X_1,\ldots,X_n$ be i.i.d. random variables with distribution $F$. It is known that if $F$ is the standard normal distribution then $$ S:=\sum_{i=1}^n X_i^2 $$ has a chi square distribution with $n$ degrees of freedom.
I remeber that the converse is an open problem: if $S$ has a chi square distribution with $n$ degrees of freedom then $F$ has to be the standard normal.
Do you have some references on this problem? (I remember that some instances has been solved, but I couldn't find them anymore)