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tup
10-22-2010, 10:04 AM
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

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?

BGM
10-22-2010, 12:01 PM
1) I am not sure if the relationship holds both way:

X_n \xrightarrow{P} 0 \iff
\forall \epsilon > 0, \lim_{n\to\infty} \Pr\{|X_n - 0| \geq \epsilon\}
= \lim_{n\to\infty} \Pr\{X_n = 1\} = \lim_{n\to\infty} p_n = 0

If \epsilon > 1 , then obviously \Pr\{X_n \geq \epsilon\} = 0 ~ \forall n

tup
10-22-2010, 12:39 PM
Thank you BGM. I guess I can use ur way for --> and I use Chebyshev for these other direction.