# Finding the limit of a product of random variables.

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.

• We can compute the distribution of $\prod_{i=1}^n X_i$ (math.stackexchange.com/questions/659254/…) and from there the distribution of $\left(\prod_{i=1}^n X_i\right)^{1/n}$. But your approach is simpler. – Math1000 May 26 '18 at 19:50
• @Math1000 Thank you for your reply. Did you find any mistake in my method ? – student_R123 May 26 '18 at 20:24
• Your solution is perfectly fine. – Sangchul Lee May 27 '18 at 3:34
• @SangchulLee Thank you. – student_R123 May 27 '18 at 15:33