[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]
(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 Borel-Cantelli Lemma. What should i do for the other direction?
Thank you in advanced!!!
Oct 22nd 2010, 12:13 PM
For ii) the reverse way, use Borel-Cantelli 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 Borel-Cantelli lemma on the internet ! :D