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Math Help - Convergence of random variables

  1. #1
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    Convergence of random variables

    Given a sequence of independent random variables (w_i)_i with non-negative values, how do you prove that:

    the series \Sigma w_i converges almost surely iif the expectation series \Sigma \mathbb{E}(w_i/(1+w_i)) converges.

    the \Rightarrow implication is easy but I'm not certain about the other way round. When \Sigma \mathbb{E}(w_i/(1+w_i)) converges, using Markov inequality and the Borel-Cantelli lemma, one can prove that w_i \rightarrow 0 almost surely, hence finite expectation and variance for i large enough (This relies on the fact the function x \rightarrow x/(1+x) is strictly increasing on [0,\infty[). But how do we know about the series behavior?

    I thought about using Cauchy's criteria with Kolmogorov inequality, in a similar way to the 3 series Theorem, but I'm probably wrong. I suspect there is a much simpler way of doing, I would really appreciate an expert view on this one.

    Thanks by advance for your help.
    Last edited by akbar; November 17th 2009 at 05:34 PM. Reason: put the mathematical parts in TeX form for clarity.
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