# Second Moments: Convergence in Probability

• Apr 25th 2013, 05:09 AM
meeksoup
Second Moments: Convergence in Probability
Hi,

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

Thanks.
• Apr 25th 2013, 05:29 PM
chiro
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.
• Apr 26th 2013, 02:34 PM
meeksoup
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.
• Apr 26th 2013, 05:08 PM
chiro
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].
• Apr 27th 2013, 12:53 AM
meeksoup
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.
• Apr 27th 2013, 01:15 AM
chiro
Re: Second Moments: Convergence in Probability
Yeah that's basically it.