[SOLVED] Some problems (limsup, liminf, ...)

Hi ^^

Probability again ! :D

So now we're dealing with Borel-Cantelli's lemma, limsup and liminf, and associates... And this is desperating to see how it is difficult to grasp the concept :D

We'll correct these exercises next week and we don't have to solve them before, but I'm not kind of patient (Rofl) I tried to do them today, but nothing came out from these ones... Just getting hints would really help me go through... ^^

By definition, $\displaystyle \liminf_n A_n=\bigcup_k \bigcap_{n \ge k} A_n$

$\displaystyle \limsup_n A_n=\bigcap_k \bigcup_{n \ge k} A_n$

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**1.**

$\displaystyle (X_n)$ is a sequence of iid Bernoulli(p), p $\displaystyle \in$ (0,1), random variables

- show that there is almost everywhere an infinity of n such that $\displaystyle X_n=1$

>>> So basically, I have to show that $\displaystyle \mathbb{P}(\cap \cup A_n)=1$, that is to say $\displaystyle \mathbb{P}(\limsup_n A_n)=1$, where $\displaystyle A_n=\{X_n=1\}$

I thought I could use the inequality $\displaystyle \mathbb{P}(\limsup A_n)\ge \limsup \mathbb{P}(A_n)$

But I don't see how to get the RHS go to 1... (I told you I have problems with limsup :D)

- show that for any n, if we define $\displaystyle B_n=\{X_n=X_{n+1}=\dots=X_{2n-1}=1\}$, there is, almost everywhere, only a finite number of $\displaystyle B_n$ that are realised.

>>> huh ???

**2.**

Let $\displaystyle (X_n)$ be a similar sequence as above.

Let $\displaystyle Y_n=X_nX_{n+1}$ and $\displaystyle V_n=Y_1+\dots+Y_n$

Show that $\displaystyle \frac{V_n}{n}$ converges in probability to $\displaystyle p^2$

>>> so I have to show that $\displaystyle \forall \epsilon>0,~ \mathbb{P}\left(\left|\tfrac{V_n}{n}-p^2 \right|> \epsilon\right) \longrightarrow 0$

and then... ? XD

thanks in advance ^^