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Math Help - Almost sure convergence

  1. #1
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    Almost sure convergence

    Does (X_{n})_{n_{\geq 1}}, X_{n}=\frac{n}{ln(n)}Y_{n}, where Y_{n} \sim Exp(n) converge a.s. to 0?
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  2. #2
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    Quote Originally Posted by yavanna View Post
    Does (X_{n})_{n_{\geq 1}}, X_{n}=\frac{n}{ln(n)}Y_{n}, where Y_{n} \sim Exp(n) converge a.s. to 0?
    You should use Borel-Cantelli lemma. Have you tried?
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  3. #3
    MHF Contributor matheagle's Avatar
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     \sum_{n=1}^{\infty}P(nY_n\ln n>\epsilon) =\sum_{n=1}^{\infty}P(Y_n>\epsilon \ln n/n)

     =\sum_{n=1}^{\infty}\int_{\epsilon \ln n/n}^{\infty}{e^{-x/n}dx\over n}

     =\epsilon \sum_{n=1}^{\infty}{1\over n^2}<\infty for all \epsilon >0
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