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Math Help - Almost sure convergence

  1. #1
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    Almost sure convergence

    Y_n=\left\{\begin{array}{cc}0,&\mbox{ with probability }<br />
1-1/n^2 \\n, & \mbox{ with probability } 1/n^2\end{array}\right.

    Show that Y_n converges almost surely to 0.



    I dont know this problem. please can you help.

    sarah
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  2. #2
    Newbie eigenvex's Avatar
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    You can see that as n -> infinity, the probability of Y = 0 approaches 1. This is because as n gets bigger, 1/(n^2) gets smaller and smaller.
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  3. #3
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    Quote Originally Posted by eigenvex View Post
    You can see that as n -> infinity, the probability of Y = 0 approaches 1. This is because as n gets bigger, 1/(n^2) gets smaller and smaller.
    Thanks but this is obvious. how can you prove this formally?? this is not just probability convergence but almost sure convergence.
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  4. #4
    Newbie eigenvex's Avatar
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    Hmm....

    Almost sure convergence is defined as



    So it can be seen that P(lim n->inf (Yn) = 0) = 1 because lim(1/(n^2)) = 0. I'm not sure how much more you have to prove there, I think all you should have to do is plug in what Yn is to that equation and show what happens as n->inf.
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  5. #5
    MHF Contributor matheagle's Avatar
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    you have something stronger here
    you have complete convergence
    Complete Convergence and the Law of Large Numbers
    which via Borel-Cantelli implies a.s. convergence
    http://www.ncbi.nlm.nih.gov/pmc/arti...01695-0003.pdf
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  6. #6
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    Quote Originally Posted by eigenvex View Post
    You can see that as n -> infinity, the probability of Y = 0 approaches 1. This is because as n gets bigger, 1/(n^2) gets smaller and smaller.
    This proves convergence in probability, not almost-sure.

    If you prove that, almost-surely, Y_n=0 for all n large enough, then of course this implies that almost-surely Y_n\to 0.

    Let A_n=\{Y_n\neq 0\}. Prove that Borel-Cantelli lemma applies to the sequence of events (A_n)_n. This will imply what I said above.
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