leo nidas

06-17-2010, 12:42 PM

I have a basic question concerning the central limit theorem

let the 2 by one vectors x1, x2, ..., xn come from a population with mean μ=[μ1 μ2]' , and variance Σ=[σ11 σ12;σ21 σ22].

Τhe clt states that sqrt(n)(Xbar-μ)~Ν(0,Σ) where Xbar=(x1+x2+...+xn)/n = (Σx(i))/n and because I had a problem typing in latex the tilde "~" will be "converge in distribution" throughout this post.

So, can I say the following?:

1. sqrt(n)(Xbar-μ)~Ν(0,Σ) =>

2. (Xbar-μ)~Ν(0,Σ/n) =>

3. Xbar~N(μ, Σ/n) =>

4. (Σx(i))/n~N(μ, Σ/n) =>

5. Σx(i)~N(μ, nΣ).

So are the above 5 steps correct? Can I claim that the sum at last step has approximately that normal? I think there is a problem with that "n" in the variance. As n goes to infinity then the variance goes to infinity? Then we have no convergence i think?! Where is my mistake?

Generally can I say somthing about the distribution of that sum?

Thanx in advance for any answers!!

let the 2 by one vectors x1, x2, ..., xn come from a population with mean μ=[μ1 μ2]' , and variance Σ=[σ11 σ12;σ21 σ22].

Τhe clt states that sqrt(n)(Xbar-μ)~Ν(0,Σ) where Xbar=(x1+x2+...+xn)/n = (Σx(i))/n and because I had a problem typing in latex the tilde "~" will be "converge in distribution" throughout this post.

So, can I say the following?:

1. sqrt(n)(Xbar-μ)~Ν(0,Σ) =>

2. (Xbar-μ)~Ν(0,Σ/n) =>

3. Xbar~N(μ, Σ/n) =>

4. (Σx(i))/n~N(μ, Σ/n) =>

5. Σx(i)~N(μ, nΣ).

So are the above 5 steps correct? Can I claim that the sum at last step has approximately that normal? I think there is a problem with that "n" in the variance. As n goes to infinity then the variance goes to infinity? Then we have no convergence i think?! Where is my mistake?

Generally can I say somthing about the distribution of that sum?

Thanx in advance for any answers!!