# converges in prob

• Oct 22nd 2010, 07:57 AM
tup
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 Borel-Cantelli Lemma. What should i do for the other direction?

Thank you in advanced!!!
• Oct 22nd 2010, 01:13 PM
Moo
Hello,

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

And to answer your first question, have a look here : http://www.mathhelpforum.com/math-he...ty-116269.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 !