Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as
$$
p_X(x) =
\left\{
\begin{array}{ll}
\frac{1}{2}\alpha L^\alpha |x|^{-\alpha-1} & |x| \geq L \\
0 & \mbox{otherwise.}
\end{array}
\right.
$$

If possible, I would like to derive PDF $p_S(x)$, where $S = \sum_{i=1}^n X_i$. Each $X_i$ is i.i.d. according to the PDF given above. I’m particularly interested in the tail distribution $\overline{F}_S(x) = Pr(S > x)$.

My approach is to calculate the $n$-th power of the CF $\varphi_X(\omega)$ and calculate its Fourier transform to obtain $p_S(x)$. However, I’m uncertain if this procedure is valid here. Is it? Do any restrictions emerge inevitably w.r.t. $\alpha$? I observe some “unhandy terms”. Is it possible to obtain $p_S(x)$ analytically and in closed form?

First of all, the CF of a (one sided) Pareto distribution is given by $\varphi_Y(\omega) = \alpha L^\alpha (-i\omega)^\alpha\Gamma(-\alpha, -i\omega L)$. Is the upper incomplete gamma function well defined for a real negative exponent and imaginary integral bounds?

I have
\begin{eqnarray}
\varphi_S(\omega) &=& \frac{1}{2^n}(\varphi_Y(\omega) + \varphi_Y(-\omega))^n\\
&=& \left(\alpha L^\alpha\right)^n|\omega|^{\alpha n}|\Gamma(-\alpha, i\omega L)|^{n} \cos \left( \arg \left( \Gamma(-\alpha, i\omega L)\right)+ \sigma(\omega)\frac{\alpha\pi}{2}\right)\;,
\end{eqnarray}
where $\sigma(\cdot)$ is the signum function.

Alternatively, I have
\begin{eqnarray}
\varphi_S(\omega) &=& \left( \alpha L^\alpha \int\limits_{L}^\infty x^{-\alpha-1} \cos \left(\omega x \right) \,\mathrm{d}x \right)^n.
\end{eqnarray}

Does this help in any way to get $p_S(x)$ or $\overline{F}_S(x)$? How could I proceed?

P.S.: In my case, n is a power of 2. If the inversion has no chance to succeed for a fix $n$, what would be an appropriate approach to investigate $\lim_{n \to \infty} \overline{F}_S(x)$?

Solutions Collecting From Web of "Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables"