Sum of n independent F distribution random variables [closed]

I need a help:

What will be the distribution of sum of $n$ independent F distributed random variables with parameters 1 and 1 (i.e., $F(x;1,1)$?

Formally, say $x_1,\ldots,x_n$ are i.i.d. as F(1,1), what is the distribution of $\sum x_i$?

Great if you can suggest some references too, Thanks in advance.

closed as unclear what you're asking by Denis-Charles Cisinski, Denis Serre, Neil Strickland, Stefan Kohl, Karl SchwedeOct 14 '14 at 17:36

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

I doubt that there is a closed form in general. The characteristic function of one of your random variables is, according to Maple, ${{\rm e}^{-is}}{\rm erfc} \left( \sqrt {-is} \right)$, so the characteristic function of your sum is ${{\rm e}^{-ins} }{\rm erfc} \left( \sqrt {-is} \right)^n$. For $n=2$ the PDF for $x > 0$ is $$\dfrac{2}{\pi (2+x)\sqrt{1+x}}$$ For $n=3$ it is $${\frac {2}{{\pi }^{2} \left( 3+x \right) \sqrt {2+x}} \left( \arctan \left({\frac {{x}^{2}+2\,x-1}{2\;\sqrt {2x+x^2}}} \right) -\arctan \left( {\frac {1}{\sqrt {2x+x^2}}} \right) + \pi \right) }$$