Sum of Lomax random variables

Question

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. I am trying to solve it by Laplace transform. Defining $f_n(x)$ as the pdf of $S_n$ and $F_n(s)$ as its Laplace transform, we have$$F_n(s)=(\mathcal{L}\{f(t)\}(s))^n=(me^{s}E_{m+1}(s))^n,$$ where $E_{m+1}(s)=\int_{1}^{\infty}\frac{e^{-st}}{t^{m+1}}dt$ is the generalized exponential integral. The next step is to take the inverse Laplace transform in terms of $F_n(s)$, that is, $$f_n(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}e^{st}(me^{s}E_{m+1}(s))^nds.$$ So, question is how to solve the above inverse Laplace transform. Other methods to determine or approximate the distribution of $S_n$ will also be appreciated.

Answer 1

This problem has been solved by CM Ramsay, etc. See here for further references.

I give the main result (the pdf) from this paper, which can be found in the page $401$: \begin{equation} f_{n}(t)=\frac{1}{n \beta} \int_{0}^{\infty} e^{-\left(1+\frac{t}{n \beta}\right) v} \varphi_{m, n}(v / n) d v \end{equation} where \begin{equation} \varphi_{m, n}(v)=(-1)^{n+1} m^{n} \sum_{r=0}^{\lfloor(n-1) / 2\rfloor}\left(-\pi^{2}\right)^{r}\left(\begin{array}{c} n \\ 2 r+1 \end{array}\right)\left(\mathrm{Ei}_{m+1}(v)\right)^{n-2 r-1}\left(\frac{v^{m}}{m !}\right)^{2 r+1} \end{equation}