The PDF of the sum of "L" non central chi-square random variables?

Question

i need to find the PDf of the summation of limited number of independent, iid, non-central chi-square random variables. I found out that there is a PDF formula for almost all the sums of random variables, except the non-central chi-square $RV_s$, so anyone knows what it may be?

Answer 1

The non-central chi-squared distribution has characteristic function $$\varphi_{k,\,\lambda}\left( t\right):=\left( 1-2it\right)^{-\tfrac{k}{2}}\exp\frac{i\lambda t}{1-2it}.$$ Summing $n$ iids with this distribution obtains the characteristic function $\varphi_{k,\,\lambda}^n=\varphi_{kn,\,\lambda n}$. The pdf is therefore $$\dfrac{1}{2}e^{-\tfrac{x+n\lambda}{2}}\left(\dfrac{x}{n\lambda}\right)^{\tfrac{kn}{4}-\tfrac{1}{2}}I_{\tfrac{kn}{2}-1}\left( \sqrt{n\lambda x}\right).$$