# $\sum_i X_i^2$ has $\chi^2_{n}$ distribution and $X_i$ i.i.d. imply $X_i$ normal

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)

• Have you looked in Kagan, Linnik, and Rao, Characterization problems in mathematical statistics? – kimchi lover Nov 3 '18 at 17:57
• Surely this "open problem" is trivial because folded normal $X_i$ would work as well. I think you mean $X_i^2$ is conjectured to be $\chi_1^2$. – J.G. Nov 3 '18 at 17:59
• @kimchilover In p.466 Section "unsolved problem" you can find the following: "Let $X_1,\ldots,X_n$ be i.i.d. r.v.'s. What can be said about the distribution of $X_1$ if it is known that $\sum X_i^2$ has a chi-square distribution, that $\sum (X_i-\overline{X})^2$ has a chi-square distribution." – Paolo Leonetti Nov 3 '18 at 18:09
• Here you can find the (positive) solution for the case of infinitely divisible random variables: tandfonline.com/doi/abs/10.1080/… – Paolo Leonetti Nov 3 '18 at 18:20