# Math Help - Convergence in Probability

1. ## Convergence in Probability

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!

2. ## Re: Convergence in Probability

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.

3. ## Re: Convergence in Probability

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