Limit distribution of $\sqrt{n}(Y_n-e)$

Question

I'm trying to find limit distribution of $\sqrt{n}(Y_n-e)$ as $n \to\infty$, where:

$$Y_n=\left(\prod_{i=1}^n U_i\right)^{-1/n}$$ and $U_i$ is i.i.d. uniform distributions in interval $(0,1)$.

My thinking so far is to use formula for uniform distribution product:

$$P_{U_1\cdots U_n}(u)=\frac{(-1)^{n-1}}{(n-1)!}(\ln u)^{n-1}$$

and then by using limit rules get the final solution.

However, I feel that the path I have chosen is very complicated and I wonder is there a more elegant solution?

Answer 1

The central limit theorem will give you the limit distribution for $\sqrt n ( \log Y_n - d)$ for some $d$ that depends on the distribution of $\log U_n$. Then apply the ``delta method'' (to the function $x\mapsto \exp(x)$) to finish the job.