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I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Levy "Sur une application de la dérivée d’ordre non entier au calcul des probabilités" page 1118 of CRAS 1923.

It might be good to also give the gist of Levy's very simple argument: if the Fourier transform was positive then the exponent would be the log-moment generating function of a random variable $X$. But if $m>1$, the second derivative at the origin vanishes and thus also the variance of $X$. The positive function would be a delta function which is a contradiction.

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Lévy "Sur une application de la dérivée d’ordre non entier au calcul des probabilités" page 1118 of CRAS 1923.

It might be good to also give the gist of Levy's very simple argument: if the Fourier transform was positive then the exponent would be the log-moment generating function of a random variable $X$. But if $m>1$, the second derivative at the origin vanishes and thus also the variance of $X$. The positive function would be a delta function which is a contradiction.

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Levy "Sur une application de la dérivée d’ordre non entier au calcul des probabilités" page 1118 of CRAS 1923.

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Levy "Sur une application de la dérivée d’ordre non entier au calcul des probabilités" page 1118 of CRAS 1923.

It might be good to also give the gist of Levy's very simple argument: if the Fourier transform was positive then the exponent would be the log-moment generating function of a random variable $X$. But if $m>1$, the second derivative at the origin vanishes and thus also the variance of $X$. The positive function would be a delta function which is a contradiction.

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor.

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Levy "Sur une application de la dérivée d’ordre non entier au calcul des probabilités" page 1118 of CRAS 1923.