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I am presented with the following Characteristic Function $$ \phi(\xi) = e^{-2\vert \xi \vert^{1.5} - 0.3 \xi^2 + 1.5i\xi} $$ corresponding to some random variable, and I am tasked with finding the specific random variable as a sum of independent random variables. At first, I thought this problem was quite simple. We may represent the provided characteristic function as $$ \phi(\xi) = e^{-2\vert \xi \vert^{1.5} - 0.3 \xi^2 + 1.5i\xi} = \left( e^{-2\vert \xi \vert^{1.5}} \right) \left( e^{1.5i\xi - 0.3 \xi^2} \right) $$ and, noting that for two independent random variables $X_1$ and $X_2$ the characteristic function of the scaled sum $aX_1 + bX_2$ takes the form $$ \phi_{aX_1 + bX_2}(\xi) = \phi_{X_1}(a\xi) \phi_{X_2}(b\xi) $$ we may then directly identify $X_2 \sim \text{N}(\mu = 1.5, \sigma^2 = 0.6)$. This is where I encounter difficulty, I am unable to identify $X_1$ as some well-known random variable. At first I thought it was corresponding to a Cauchy$(\xi_0, \gamma)$ random variable with $\xi_0 = 0$ and $\gamma = -2$, but the Characteristic Function of such random variable takes form $$ \phi_{\text{Cauchy}(\xi_0, \gamma)} = e^{-2\vert \xi \vert} $$ However, this does not match the power of $1.5$ on $\vert \xi \vert$ in the original $\phi(\xi)$.

Is there some well-known random variable with such Characteristic Function, or perhaps an algebraic process to reveal such a random variable? Any direction is well appreciated!

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A symmetric $\alpha$-stable distribution has characteristic function $$\varphi(\xi)=\exp(-|c\xi|^\alpha).$$ For further reading, see here

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