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Math Help - Expectation in probability convergence

  1. #1
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    Expectation in probability convergence

    Prove that X_n converges to 0 in probability if and only if  E ( \frac { |X_n | } {1 + |X_n|}) \rightarrow 0

    So I know that  P(|X_n-0| \geq \epsilon ) \rightarrow 0 , but I don't really know how I should rewrite the expectation value part to make this work. Thank you!
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    Super Member girdav's Avatar
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    Re: Expectation in probability convergence

    You have to use the fact that the map x\mapsto\frac x{x+1} is increasing. If we assume that E\left[\frac{|X_n|}{1+|X_n|}\right]\to 0, for a fixed \varepsilon, \frac{\varepsilon}{1+\varepsilon} P(|X_n|\geq \varepsilon) can be bounded above by E\left[\frac{|X_n|}{1+|X_n|}\right].
    Thanks from tttcomrader
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