Convergence in Probability

**LHeiner** · October 4th 2011, 10:06 PM

Hi

how do I proof the following lemma:

$X_1,X_2,X_3,\cdots$ independent r.v. with $\mathbb{P}(X_n=1)=p_n$ and $\mathbb{P}(X_n=0)=1-p_n$ .

1) $X_n \overset{P}{\rightarrow} X \Leftrightarrow p_n \rightarrow 0$
2) $X_n \rightarrow X a.s. \Leftrightarrow \sum_{n=1}^\infty p_n <\infty$

thank you!

**girdav** · October 5th 2011, 10:21 AM

For the first problem, compute $P\left(|X_n-X_m|\geq \frac 12\right)$ .
If we assume that the sequence converges in probability then the limit when $m,n\to\infty$ will be $0$ , and it show that the limit of the sequence $\{p_n\}$ is $0$ (the converse will follow from the previous computation).
For the second problem, use Borel-Cantelli lemma.
It's an interesting exercise, since we can easily give an example of a sequence of random variables which converges in probability but not almost everywhere.

**matheagle** · October 8th 2011, 08:04 PM

Well it depends, I assume X=0, then $p_n\to 0$
But X can be 1 and then $p_n\to 1$

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