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Math Help - Second Moments: Convergence in Probability

  1. #1
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    Second Moments: Convergence in Probability

    Hi,

    I wish to get feedback as to whether my answer to the following is correct/adequate.

    Attached for your review.

    Thanks.
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  2. #2
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    Re: Second Moments: Convergence in Probability

    Hey meeksoup.

    The critical step in the proof will be that the Var(1/n Sigma_k X_k^2) = 1/n^2 * nk = k/n. Since k is fixed, we take the limit as n -> infinity and get the variance to be 0 which means its consistent.

    You need to specify this and make it clear otherwise you probably won't get the marks.
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  3. #3
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    Re: Second Moments: Convergence in Probability

    Thanks chiro,

    could you please explain how the variance of the sum of second moments is equal to nk or direct me to some resources?

    this is the part where I am having troubles in evaluating.

    Thanks.
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  4. #4
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    Re: Second Moments: Convergence in Probability

    Basically the rule is Var[X + Y] = Var[X] + Var[Y] if X and Y are independent and E[X_i^2] = Var[X_i].
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  5. #5
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    Re: Second Moments: Convergence in Probability

    so basically its (1/n^2) times the n number of Var(X_i). the n cancel out, leaving the var(X_i) divide by n. as n approaches infinity, the whole thing approaches zero.
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  6. #6
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    Re: Second Moments: Convergence in Probability

    Yeah that's basically it.
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