[SOLVED] Convergence in Probability

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February 21st 2010, 08:34 PM
Statistik

[SOLVED] Convergence in Probability

Hi,

I'm trying to show that in probability

$ \sqrt {n} * [\overline X_n - EX ] \rightarrow 0 $

*** 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) ?

$ \sqrt {\frac {1}{n}} * \sum_{i=1}^n (x_i - EX_i ) \rightarrow 0 $

Thanks!
February 21st 2010, 10:53 PM
matheagle

Quote:

Originally Posted by Statistik

Hi,

I'm trying to show that in probability

$ \sqrt {n} * [\overline X_n - EX ] \rightarrow 0 $

*** 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) ?

$ \sqrt {\frac {1}{n}} * \sum_{i=1}^n (x_i - EX_i ) \rightarrow 0 $

Thanks!

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

Usually

$ [\overline X_n - EX ] \rightarrow 0 $

so we need additional conditions to make

$ \sqrt {n} * [\overline X_n - EX ] \rightarrow 0 $

It would be easier if you asked for

$ {[\overline X_n - EX ]\over \sqrt n} \rightarrow 0 $
March 2nd 2010, 08:56 PM
Statistik

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!