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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)

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  • $\begingroup$ Have you looked in Kagan, Linnik, and Rao, Characterization problems in mathematical statistics? $\endgroup$ – kimchi lover Nov 3 '18 at 17:57
  • $\begingroup$ 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 $\chi_1^2$. $\endgroup$ – J.G. Nov 3 '18 at 17:59
  • $\begingroup$ @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." $\endgroup$ – Paolo Leonetti Nov 3 '18 at 18:09
  • $\begingroup$ Here you can find the (positive) solution for the case of infinitely divisible random variables: tandfonline.com/doi/abs/10.1080/… $\endgroup$ – Paolo Leonetti Nov 3 '18 at 18:20

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