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Random variables $(X_{n})$ are independent and $X_{n}$ has an uniform distribution on $[0,n]$ for n=1,2,... Prove that:

$$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}\rightarrow \infty$$ almost sure.

We can write $X_{n}=nU_{n}$ where $U_{n}$ are iid and have an uniform distribution on [0,1].

Then from LLN we have:

$$\overline{U_{n}} \rightarrow \text{E}U_{1}=\frac{1}{2}$$.

Is it a good approach? Can I use this result to solve the main problem?

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Hints:

  1. By the strong law of large numbers, $$\frac{X_1+ \frac{X_2}{2}+\ldots+ \frac{X_n}{n}}{n} \to \frac{1}{2}$$ almost surely. Show that this implies $$\frac{\frac{X_k}{k}+ \ldots+ \frac{X_n}{n}}{n} \to \frac{1}{2}$$ almost surely for any (fixed) $k \in \mathbb{N}$.
  2. Using the the non-negativity of the random variables, show that $$\frac{1}{k} \frac{X_1+\ldots+X_n}{n} \geq \frac{1}{k} \frac{X_k+\ldots+X_n}{n} \geq \frac{\frac{X_k}{k}+ \ldots+ \frac{X_n}{n}}{n}.$$
  3. Combining step 1 and 2 conclude that $$\liminf_{n \to \infty} \frac{X_1+\ldots+X_n}{n} \geq \frac{k}{2}.$$
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  • $\begingroup$ $$\liminf_{n \to \infty} \frac{1}{k}\frac{X_{1}+X_{2}+\dots+X_{n}}{n} \ge \liminf_{n \to \infty} \frac{\frac{X_k}{k}+ \ldots+ \frac{X_n}{n}}{n}= \frac{1}{2}$$ $\endgroup$ – mrnobody Jun 13 '15 at 20:14
  • $\begingroup$ thanks a lot @saz ;) Your hints are really helpful.. $\endgroup$ – mrnobody Jun 13 '15 at 20:15
  • $\begingroup$ @sebuss007 Thanks, you are welcome. $\endgroup$ – saz Jun 14 '15 at 5:27
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Does this hint help: $$var(\bar{X})=\frac{1}{n^2}\sum_{j=1}^{n}\frac{1}{12}j^2$$ hence $var(\bar{X})\to \infty$ and $n \to \infty$.

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  • $\begingroup$ Sorry, I don't know how I can use this hint... $\endgroup$ – mrnobody Jun 13 '15 at 19:08

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