
converges in prob
[tex]Let X_1, X_2, ... be independent with P(X_n = 1) = p_n and P(X_n = 0) = 1 p_n. Show that (i) X_n \rightarrow 0 in probability iff p_n \rightarrow 0, and (ii) X_n \rightarrow 0 a.s iff \sum p_n \rightarrow \infty[\MATH]
Attemp
(i) I used Chebyshev's ineq for X_n \rightarrow 0 in probability implies p_n \rightarrow 0. Can someone give hint for the other direction?
(ii)\sum p_n \rightarrow \inft implies X_n \rightarrow 0 a.s by BorelCantelli Lemma. What should i do for the other direction?
Thank you in advanced!!!

Hello,
For ii) the reverse way, use BorelCantelli lemma 2, since you have the independence of the random variables.
If you don't know it, just find a proof for the second part of BorelCantelli lemma on the internet ! :D
And to answer your first question, have a look here : http://www.mathhelpforum.com/mathhe...ty116269.html
it may be helpful for you to read the thread, and you may find the way to prove i) : there is no need to use Chebyshev !