Approximate distribution of product of N normal i.i.d.? Special case μ≈0 Given 
$N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, 
and $\mu_X \approx 0$,
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 $\mu_X \approx 0$ of a more general question.
 A: It is possible to obtain an exact solution in the zero-mean case (part B).
The Problem 
Let $(X_1, \dots, X_n)$ denote $n$ iid $N(0,\sigma^2)$ variables, each with common  pdf $f(x)$:

We seek the pdf of $\prod_{i=1}^n X_i$, for $n = 2, 3, \dots$
Solution
The pdf of the product of two such Normals is simply:

... where I am using the  TransformProduct function from the mathStatica package for Mathematica. The domain of support is:

The product of 3, 4, 5 and 6 Normals is obtained by iteratively applying the same function (here four times):

... where MeijerG denotes the Meijer G function
By induction, the pdf of the product of $n$ iid $N(0,\sigma^2)$ random variables is:

$$\frac{1}{(2 \pi )^{\frac{n}{2}} \sigma ^n} \text{MeijerG}[\{ \{ \},  \{ \} \}, \{ \{0_1, \dots, 0_n \}, \{ \} \}, \frac{x^2}{2^n \sigma ^{2 n}}] \quad \quad \text{ for } x \in \mathbb{R} $$

Quick Monte Carlo check
Here is a quick check comparing:


*

*the theoretical pdf just obtained (when $n = 6$ and $\sigma=3$):  RED DASHED curve

*to the empirical Monte Carlo pdf:  squiggly BLUE curve



Looks fine! [ the blue squiggly Monte curve is obscuring the exact red-dashed curve ]
