# Can we approximate the distribution of S?

I want to understand how the sampling distribution of the whole covariance matrix behaves for large $n$. I am trying to use the delta method and multivariate CLT. I am trying to show that when the $X$'s are IID with nonzero finite fourth order moments then the operation (sample variance) takes place, as in this equation:

$$\sqrt{n}(s_X^2 - \sigma^2) \underset{n\rightarrow \infty}{\overset{d}{\longrightarrow}} N(0, \mathbb{E}X_1^4 - (\mathbb{E}X_1^2)^2).$$

(For simplicity I have assumed $\mathbb{E}X_i = 0$). Help is solicited in understanding the steps I need to take. I started with studying the asymptotic behavior of the sample variance and then went on to use the delta method for the following function $g(a,b) = b - a^2$.

• Here's the link of the result I'd like to get to...imgur.com/kJxIR
– ray
Commented Dec 2, 2012 at 11:12
• Why do you want to use the multivariate CLT and the delta method here? You can get by with a simpler analysis. Is your aim just to practice the techniques you've mentioned or are you more interested in simply knowing an efficient and economical means of obtaining the stated result? Commented Dec 3, 2012 at 23:13
• I'm open to other methods as well. It's just that my course of study deals with these tools of analysis so I was working my way thru them. I still am and I think i'm some way off getting to my solution...
– ray
Commented Dec 4, 2012 at 15:27

## 1 Answer

To close this one, this snapshot is from Dasgupta's 2008, Asymptotic Theory of Statistics and Probability, ch 3, p.41-42:

"$X_i$'s as above" means i.i.d.