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?


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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 $e^{\log (X_1 \ldots X_n)^{\frac{1}{n}}} = e^{\frac{1}{n}\sum \log X_n }$.


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.

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

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