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Is there any intuitive reason of the independence assumption behind the second part of the Borel-Cantelli lemma. or it is just for calculation ?

aaaaaa
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In the first part you don't really need it because the sum is converging already so you are not building the sum over the same events all the time!

If you do the second part with non-independent events you might just build the sum over the same event all the time, so your sum becomes infinite. But that does not mean that the Probability is equal to 1.

Edit:

Ok fail! Sorry! The Independence assumption can be weakened. There are a lot different forms with weaker assumptions. But this paper claims that you do not need any assumption for the second part at all.
This totally amazed me there's also another post here already with a proof and a weaker assumption Generalized Second Borel Cantelli Lemma.

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mjb4
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  • Is there any confusion between "independent" and "disjoint" events here. looks like so – aaaaaa Jan 24 '14 at 00:29
  • Hmm, it's true I did really put it like I would refer to disjoint. But no I meant independent!

    Let me see if I can come up with an example but it's to late now!

     – mjb4 Jan 24 '14 at 01:29
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    The paper you linked is obviously wrong. Take $X$ a single uniform $[0,1]$ random variable and let $A_n = {X \in [0, \frac{1}{n}]}$. – Chris Janjigian Jan 24 '14 at 16:59