Multiplication of normal distributions Suppose X_1, ..., X_k are iid standard Gaussian variables, for some k > 1.

Then, what is the distribution of X := X_1 * ... * X_k ? Can it be approximated by a Gaussian, maybe for large k ?
The distribution of the multiplication of independent Gaussians is not close to a Gaussian.

There are various ways to convince yourself in this. E.g., compute the expectation and the variance of the product from the expectation and the variance of the individual variables.
This gives you a candidate Gaussian if there were a Gaussian approximation.
Now, think of some simple properties of the product (e.g., the probability it is 0) and see if your candidate is good enough as an approximation.