# Sum of two i.i.d R.V having singly non-central F distribution

Noncentral F-distribution is used frequently in communication areas. In one of the applications, I need to do a sum of two i.i.d R.V having non-central F-distribution with parameter 1 (d.o.f for numerator), $$N-1$$ (d.o.f for the denominator) and $$\lambda$$ (non-centrality) parameter of the numerator. Is there any standard result on the sum of these two R.V.? or is there any approximation result? Since applying a straightforward approach, i.e., the convolution formula, is tough or doesn't seem to have a closed-form solution.

I think you end up having a non-standard distribution. My (tentative) approach would be the following. Let $$W_{j}\sim \mathrm{F}_{1,N-1,\lambda},j=1,2$$.
First, we know that $$\widetilde{T}_j\sim \mathrm{t}_{N-1,\lambda},\widetilde{T}_j:=\sqrt{W_j},j=1,2,$$ where $$\mathrm{t}_{\nu,\mu}$$ is the non-central $$\mathrm{t}$$ distribution with $$\nu$$ degrees of freedom and non-centrality parameter $$\lambda$$.
Second, I would consider the centralized version of the two random variables above. Namely, $$T_j = \widetilde{T}_j-\frac{\lambda}{\sqrt{V/(N-1)}},\quad V\sim \chi_{N-1}^2.$$
Finally, this paper provides a formula for the density of two independent $$d$$-dimensional Student-$$t$$ random vectors. In your case $$d=1$$. If you can work with $$(T_1,T_2)$$ instead of $$(W_1,W_2)$$ you are done. Otherwise, you simply transform the density for $$T_1+T_2$$ given in the paper into the density of $$W_1+W_2$$ using the relationships mentioned above.