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Math Help - [SOLVED] Convergence in Probability

  1. #1
    Junior Member
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    [SOLVED] Convergence in Probability

    Hi,

    I'm trying to show that in probability

    <br />
\sqrt {n} * [\overline X_n - EX ] \rightarrow 0<br />

    *** Edit: See note below - just not sure how to proceed. ***

    Would it be correct to manipulate the function to get to the following, to eliminate the sqrt(n) ?

    <br />
\sqrt {\frac {1}{n}} * \sum_{i=1}^n (x_i - EX_i ) \rightarrow 0<br />

    Thanks!
    Last edited by Statistik; February 21st 2010 at 08:41 PM. Reason: On second thought, that just doesn't convert either, does it. Just not sure what to do...
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Statistik View Post
    Hi,

    I'm trying to show that in probability

    <br />
\sqrt {n} * [\overline X_n - EX ] \rightarrow 0<br />

    *** Edit: See note below - just not sure how to proceed. ***

    Would it be correct to manipulate the function to get to the following, to eliminate the sqrt(n) ?

    <br />
\sqrt {\frac {1}{n}} * \sum_{i=1}^n (x_i - EX_i ) \rightarrow 0<br />

    Thanks!

    They are the same.
    But I need more info in order to establish any convergence here.

    Usually

    <br />
 [\overline X_n - EX ] \rightarrow 0<br />

    so we need additional conditions to make

    <br />
\sqrt {n} * [\overline X_n - EX ] \rightarrow 0<br />

    It would be easier if you asked for

    <br />
{[\overline X_n - EX ]\over \sqrt n} \rightarrow 0<br />
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  3. #3
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    matheagle,

    Sorry about the hiatus. Thanks for the response... Turns out I was trying to separate parts of a problem that I should have left together (and I would have gotten to a multi-variate expression of the problem that is similar to your proposition of "it would be easier if you asked for ...").

    Thanks again!
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