I have following question and I need to know whether I did it correctly.
Let $ X_1, \cdots, X_n$ be i.i.d. Uniform ~ $(0,1)$. I need to prove that $\lim_{n\to \infty} \left(X_1 X_2 \cdots X_n\right)^{1/n}$ have a limit w.p 1 and find that limit.
This is what I did so far.
Let $Y = \left(X_1 X_2 \cdots X_n\right)^{1/n}$. Then $\log(Y) = \frac{1}{n} \sum_{i=1}^n \log(X_i)$.
Using Strong law of large numbers, $\frac{1}{n}\sum_{i=1}^n \log(X_i)$ converges a.s to its mean. (say $\mu$).
To find $\mu$, I took that $E(\log(X_1)) = -1$. (by transformation and taking expectation).
So $\log(Y) = \frac{1}{n} \sum_{i=1}^n \log(X_i)$. converges a.s to $-1$.
Therefore $Y$ converges a.s to $ e^{-1}$.
Is this correct? Are there any efficient ways of doing this proof other than this way?
Thank you.
