A note on Linnik's distribution (Q912465)

The author observes that if \(S_{\alpha}\) is a symmetric stable random variable with characteristic function \(e^{-| t|^{\alpha}}\) \((0<\alpha \leq 2)\), and if \(V_{\beta}\) \((\beta >0)\) is an independent random variable with density \(\exp (-\nu^{\beta})/\Gamma (1+1/\beta),\) \(\nu >0\), then \(X=S_{\alpha}V_{\beta}^{\beta /\alpha}\) has characteristic function \[ \phi (t)=1/(1+| t|^{\alpha})^{1/\beta}. \] By virtue of this he proves the validity and unimodality of Linnik's characteristic function \(1/(1+| t|^{\alpha})\), \(0<\alpha \leq 2\).