I have X and Y univariate independent random variables, described by Gaussian mixture pdf:
$$ f_X = \sum_i^{N_X} \alpha_{X, i} N[\mu_{X, i}, \sigma^2_{X, i}] $$
$$ f_Y = \sum_i^{N_Y} \alpha_{Y, i} N[\mu_{Y, i}, \sigma^2_{Y, i}] $$
I am searching for a good approximation (maybe another Gaussian mixture) of $f_{Z}, Z=XY$, or even better for $Z=\sqrt{XY}$
In my case $N_{X}=N_{Y}$ and it is small (=3). All the $\mu$ are different but pretty similar (their difference are smaller than the $\sigma$) so $X$ and $Y$ are unimodal. In addition $\alpha_1 \gg \alpha_2 \gg \alpha_3$ and $\sigma_1 \ll \sigma_2 \ll \sigma_3$.