Approximate distribution of product of N normal i.i.d.? Special case μ>10σ, σ>0 Given 
$N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,
and $|\mu_X|\geq10\sigma_X$, $\sigma > 0$,
looking for:


*

*accurate closed form distribution approximation of
$Y_N=\prod\limits_{1}^{N}{X_n}$

*asymptotic (exponential?) approximation of same product 


This is a special case $|\mu_X|\geq10\sigma_X$, $\sigma > 0$ 
of a more general question.
 A: Note that after re-scaling form Y to F to G, we can reformulate the question:
$Z=\mathcal{N}(0,1)$,
$X=\mu_X+\sigma_XZ$,
$Y_N=\mu_X^N\prod{(1+\frac{\sigma_X}{\mu_X}Z)}$,
$Y_0=\mu_X^N$,
$\alpha=\frac{\sigma_X}{\mu_X}$, 
$F =\frac{Y_N}{Y_0}=\prod{(1+\alpha Z)}$ - factor function, 
$G=log(F)/(N\alpha)$ - growth function, 
where $\alpha N$ is scaling parameter, 
apparent from $\lim(\alpha \rightarrow 0)$.
The question then becomes to find approximation of 
$\mu_G$ and $\sigma_G$, for $G$:
$$G_N=\frac{1}{\alpha N}\sum\limits_{1}^{N} log(1+\alpha Z_n)$$
Here is an intuitive outline for possible solution - an attempt of qualitative approximation to the Monte-Carlo results.
1.1 assume $\alpha$ is small and truncated taylor series can be used:
$log(1+x) \approx x - \frac{1}{2}x^2$, 
then $G$ becomes:
$G \approx \frac{1}{\alpha N}(\sum \alpha Z - \frac{1}{2}\sum \alpha^2 Z^2)$
or 
$G \approx G_A +G_B$, 
where
$G_A=\frac{1}{N}\sum Z$, $G_B=-\frac{1}{2}\frac{\alpha}{N}\sum Z^2$
1.2 assume berry-esseen can be applied to $G_A$, 
then $G_A$ becomes:
$G_A \approx 0 + \frac{1}{\sqrt N}Z_A$, 
with $Z_A=\mathcal{N}(0, 1)$
1.3 assume $\chi^2$ can be approximated as $\mathcal{N}(N, 2N)$,
then $G_B$ becomes:
$G_B \approx -\frac{1}{2}\alpha - \frac{1}{\sqrt{2N}}\alpha Z_B$,
with $Z_B=\mathcal{N}(0, 1)$
1.4 assume $G_A$ and $G_B$ are correlated with $\rho_N=\mathrm{Cov}(G_A,G_B)$
then we finally have approximate resulting $G$:
$$
G_N \approx 
-\frac{1}{2}\alpha 
+\frac{1}{\sqrt{N}}(1+\rho_N\alpha + \frac{1}{2}\alpha^2)^{\frac{1}{2}}Z
$$
This solution is already in use (granted - with some accuracy concerns).
