1. ## Convergence in Probability

Hi

how do I proof the following lemma:

$\displaystyle$X_1,X_2,X_3,\cdots$$independent r.v. with \displaystyle \mathbb{P}(X_n=1)=p_n$$ and $\displaystyle$\mathbb{P}(X_n=0)=1-p_n$$. 1) \displaystyle X_n \overset{P}{\rightarrow} X \Leftrightarrow p_n \rightarrow 0$$
2) $\displaystyle$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 $\displaystyle P\left(|X_n-X_m|\geq \frac 12\right)$.
If we assume that the sequence converges in probability then the limit when $\displaystyle m,n\to\infty$ will be $\displaystyle 0$, and it show that the limit of the sequence $\displaystyle \{p_n\}$ is $\displaystyle 0$ (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 $\displaystyle p_n\to 0$
But X can be 1 and then $\displaystyle p_n\to 1$