if $X_1,X_2,X_3,...,X_n$ is a sample size $n$ and $i.i.d.$ of a random variable with distribution $f(x)$, and
$$Y=X_1.X_2.X_3.....X_n$$
what is the approximate distribution of $\log{Y/n}$ for large sample (high $n$)?
My annotations
$\log{Y/n}=\frac{\log(X_1)+\log(X_2)+...+\log(X_n)}{n}$
Let's call $\log(X_i)$ from $Z_i$, so we have $$\frac{Z_1+Z_2+Z_3+...+Z_n}{n}=\overline{Z}n$$
by Central limit theorem: $\frac{\overline{Z}n-\mu_z}{\sigma_z/\sqrt{n}}\rightarrow_dNormal(0,1)$
But I'm stuck here, I can not find the distribution of $\log{Y/n}$