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Math Help - Convergence in Probability

  1. #1
    Junior Member
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    Convergence in Probability

    Hi

    how do I proof the following lemma:

    $X_1,X_2,X_3,\cdots$ independent r.v. with $\mathbb{P}(X_n=1)=p_n$ and $\mathbb{P}(X_n=0)=1-p_n$.

    1) $X_n \overset{P}{\rightarrow} X \Leftrightarrow p_n \rightarrow 0$
    2) $X_n \rightarrow X a.s. \Leftrightarrow \sum_{n=1}^\infty p_n <\infty$

    thank you!
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  2. #2
    Super Member girdav's Avatar
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    Re: Convergence in Probability

    For the first problem, compute P\left(|X_n-X_m|\geq \frac 12\right).
    If we assume that the sequence converges in probability then the limit when m,n\to\infty will be 0, and it show that the limit of the sequence \{p_n\} is 0 (the converse will follow from the previous computation).
    For the second problem, use Borel-Cantelli lemma.
    It's an interesting exercise, since we can easily give an example of a sequence of random variables which converges in probability but not almost everywhere.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Re: Convergence in Probability

    Well it depends, I assume X=0, then p_n\to 0
    But X can be 1 and then p_n\to 1
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