# Express Random Variable as Sum of Independent Random Variables from Characteristic Function

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!

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