# Thread: [SOLVED] Convergence in Probability

1. ## [SOLVED] Convergence in Probability

Hi,

I'm trying to show that in probability

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

*** 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) ?

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

Thanks!

2. Originally Posted by Statistik
Hi,

I'm trying to show that in probability

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

*** 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) ?

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

Thanks!

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

Usually

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

so we need additional conditions to make

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

It would be easier if you asked for

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

3. 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!