# CDF of product of two independent non-central chi distributions

I am trying to find out the expression of the CDF of the product of two independent non-central chi distributions (not chi-squared) each with $k=2$ degrees of freedom. As I understand, non-central chi distribution (with k=2) is square root of the sum of squares of two Guassian random variables of different means and variances.

If I assume that the two Guassian variables have same means and variances, then the non-central chi distribution reduces to standard Rayleigh. It turns out that a closed form of CDF of the product of two independent Rayleigh distributions does exist (e.g. p. 56 of this book).

However, I have been unable to find a closed form of CDF of the product of the two independent non-central chi distributions. Is there a good reference that I can look into? Any help would be greatly appreciated.

-ryan

In the paper, a random variable $X$ is said to have a non-central chi distribution with 2 d.f. if its p.d.f. takes the form $$f_X(x)=\frac{x}{\sigma^2}\exp\left(-\frac{\beta^2+x^2}{2\sigma^2}\right)I_0\left(\frac{\beta x}{\sigma^2}\right),$$ where $I_\nu(z)$ is the modified Bessel function of the first kind. Let $X_1\sim\chi(\beta_1,\sigma_1^2)$ and $X_2\sim\chi(\beta_2,\sigma_2^2)$ be independent non-central chi r.v.'s with 2 d.f. The authors showed that the p.d.f. of $Y=X_1X_2$ is $$f_Y(y)=\exp\left(-\frac{\beta_1^2}{2\sigma_1^2}-\frac{\beta_2^2}{2\sigma_2^2}\right)\frac{y}{\sigma_1^2\sigma_2^2}\times\sum_{j=0}^\infty\left[\left(\frac{\beta_2}{\sqrt2\sigma_2^2}\right)^{2j}y^{j\sigma_2/\sigma_1}\sum_{i=0}^j\left(\frac{[(\beta_1\sigma_2)/(\beta_2\sigma_1)]^{i}}{i!(j-i)!}\right)^2K_{i-(j/2)}\left(\frac{y}{\sigma_1\sigma_2}\right)\right],$$ where $K_\nu(x)$ is the modified Bessel function of the second kind.