# Application - Law of large numbers [closed]

Let $X_1,X_2,...$ be independent Uniform(0,1) distributed random variables.

My first problem ist to show, that $Z_n := \frac{X_1^2+...+Xn^2}{X_1+...+X_n}$ converges almost surely and to find its limit.

My second problem is to show that $\lim_{n\to +\infty} \sqrt[n]{X_1\cdots X_n}=e^{-1}$ almost surely.

Can anybody help me here?

## closed as off-topic by Did, Namaste, Ken Duna, mrnovice, Davide GiraudoApr 26 '17 at 22:30

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Namaste, Ken Duna, mrnovice, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.

Hint: You can use continuous mapping theorem for a.s. convergence, that is

Assume $f$ is continuous, and $X_n \rightarrow X$ a.s., then $f(X_n) \rightarrow f(X)$ a.s. as well.

Back to your question, in the first question, try to divide both the numerator and denominator $n$, and use apply Law of Large Numbers to $\frac{1}{n} \sum X_n$ and $\frac{1}{n} \sum X_n^2$.

For the second question, consider $e^{\log (X_1 \ldots X_n)^{\frac{1}{n}}} = e^{\frac{1}{n}\sum \log X_n }$.

EDIT

To calculate $E\log X$ it suffices to see that $E \log X = \int_0^1\log xdx = x\log x|^1_0 - \int_0^11dx =-1$. Note that you have to argue $\lim_{x \rightarrow 0} x\log x = 0$ which can be done in many ways, like this.

• @JackD'Aurizio I think it is much easier for the asker to direclty calculate the mean based on the uniform distribution. – Ran Wang Apr 25 '17 at 16:00
• @NCh: all right, that was an ineffective and useless detour, I am removing my previous comment. – Jack D'Aurizio Apr 25 '17 at 16:20
• Alright, thanks! But how to calculate $E[log X_1]$? – dish Apr 25 '17 at 17:31
• @dish, please see the edits – Ran Wang Apr 26 '17 at 12:32