Densities of stable distributions are smooth?

Asked Feb 14 '20 at 15:38

Active Feb 14 '20 at 15:38

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Fix $1<p<2$. Assume that $\mu_p$ is a $p$-stable probability distribution on the real line. That is, a probability measure whose characteristic function is $e^{-|t|^p}$.

The book "Parametric statistical inference" by James K. Lindsey, says, on page $53$, that "stable distributions are continuous, with smooth unimodal densities".

I could not find a reference for the fact that the density is smooth. Can someone please point out such a reference? Thank you.

asked Feb 14 '20 at 15:38

uniquesolution

18,541

This is a simple consequence of the inversion formula (pdf from cf) and the Leibniz integral rule. – Gabriel Romon Feb 14 '20 at 18:32
@Gabriel Romon - I think "smooth" means $C^{\infty}$. Can you prove that the density is infinitely differentiable? – uniquesolution Feb 14 '20 at 20:07
Of course, by repeated applications of the theorem., – Gabriel Romon Feb 14 '20 at 20:46
So you claim that the Fourier transform of $e^{-|t|^p}$ is infinitely differentiable by essentially differentiating under the integral sign. This does not look right to me, because , since $1<p<2$, after three derivatives you get terms that are not locally integrable in a nbd of zero, so you cannot continue differentiating. – uniquesolution Feb 14 '20 at 21:22
The derivatives of the integrand w.r.t. $x$ have the form $Kt^n\cos(xt)e^{|t|^p}$ or $Kt^n\sin(xt)e^{|t|^p}$. How are they not integrable ? – Gabriel Romon Feb 14 '20 at 21:29
Yes, you are right. Thank you. – uniquesolution Feb 14 '20 at 21:57

Densities of stable distributions are smooth?

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