# [SOLVED] Convergence in Probability

• 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!